Inference for the limiting cluster size distribution of extreme values

The Annals of Statistics 2009, V ol. 37, No. 1, 271–310 DOI: 10.1214 /07-AOS551 c  Institute of Mathematical Statistics , 2009 INFERENCE F OR THE LIMITING CLUSTER SIZE DISTRIBUTION OF EXTREME V ALUES By Christian Y. Rober t CNAM a nd CREST, F r anc e Any limiting point pro cess for th e time normalized exceedances of high levels by a stationary sequen ce is necessarily comp ound Poiss on under appropriate long range dep endence conditions. T ypically ex- ceedances appear in clusters. The underlying P oisson points represen t the cluster p osi tions and the multiplicities correspond to the cluster sizes. In the present pap er we introduce estimators of t h e limiting cluster size probabilities, which are constructed through a recursive algorithm. W e derive estimators of th e extremal ind ex which p la ys a key role in determining the intensity of cluster p ositi ons. W e study the asymp totic prop erties of th e estimators and inv estigate th eir ﬁ- nite sample b eha vior on simulated data. 1. In tro d uction. Man y results in extreme v alue theory may b e natu- rally discussed in terms of p oin t pro cesses. T ypically , the distribu tion of extreme order statistics may b e obtained by considering th e p oint pro cess of exceedances of a high lev el. More formally , let ( X n ) b e a strictly sta- tionary sequence of r andom v ariables (r.v.s) with marginal distribu tion F . W e assum e that f or eac h τ > 0 th er e exists a sequence of lev els ( u n ( τ )) suc h that lim n →∞ n ¯ F ( u n ( τ )) = τ , wh er e ¯ F = 1 − F . It is necessary and s uf- ﬁcien t for the existence of suc h a sequence that lim x → x f ¯ F ( x ) / ¯ F ( x − ) = 1 , where x f = s u p { u : F ( u ) < 1 } (see Theorem 1.7.13 in [ 28 ]). A n atural c hoice is give n by u n ( τ ) = F ← (1 − τ /n ) , where F ← is the generalized inv erse of F , that is, F ← ( y ) = in f { x ∈ R : F ( x ) ≥ y } . The p oint pro cess of time normal- ized exceedances N ( τ ) n ( · ) is deﬁned by N ( τ ) n ( B ) = P n i =1 1 { i/n ∈ B ,X i >u n ( τ ) } for an y Borel set B ⊂ E := (0 , 1]. The ev en t that X n − k +1: n , the k th largest of X 1 , . . . , X n , do es not exceed u n ( τ ) is equiv alen t to { N ( τ ) n ( E ) < k } and the Received January 2007; revised S ep tem b er 2007. AMS 2000 subje ct classiﬁc ations. Primary 60G70, 62 E20, 62M09; s econdary 62G20, 62G32. Key wor ds and phr ases. Ext reme v alues, ex ceedance p oint pro cess es, limiting cluster size d istri but ion, extremal index, strictly stationary sequences. This is an electro nic repr in t of the original ar ticle published by the Institute of Mathematical Sta tistics in The Annals of S t atistics , 2009, V ol. 3 7 , No. 1, 2 71–310 . This reprint diﬀers fr om the origina l in pa g ination and typogr aphic detail. 1 2 C. Y. ROBER T asymptotic d istribution of X n − k +1 : n is easily d eriv ed from the asymptotic distribution of N ( τ ) n ( E ). If ( X n ) is a sequence of indep endent and id en tica lly d istributed (i.i.d.) r.v.s, N ( τ ) n con v erges in distribu tio n to a homogeneous P oisson pro cess with in tensit y τ (see, e.g., [ 13 ], Th eorem 5.3.2). If th e i.i.d. assu mption is relaxed and a long range dep endence condition is assumed [∆( u n ( τ )) deﬁn ed b elo w], the limiting p oin t pro cess is necessarily a h omog eneous comp ound P oisson pro cess with intensit y θ τ ( θ ≥ 0) and limiting cluster size distrib ution π [ 24 ]. The constan t θ is referred to as th e extremal index and its recipro cal is equal to the mean of π und er some mild additional assump tions (see [ 36 , 38 ] for some counte rexamples). It may b e sho wn that θ ≤ 1 and that the comp ound P oisson limit b ecomes Poisson when θ = 1. If lim n →∞ P ( N ( τ ) n ( E ) = 0) = e − θ τ , then a necessary and suﬃcient condi- tion for con ve rgence of N ( τ ) n is con v ergence of the conditional distribution of N ( τ ) n ( B n ) with B n = (0 , q n /n ] giv en th at there is at least one exceedance of u n ( τ ) in { 1 , . . . , q n } to π , that is, lim n →∞ P ( N ( τ ) n ( B n ) = m | N ( τ ) n ( B n ) > 0) = π ( m ) , m ≥ 1 , (1.1) where ( q n ) is a ∆( u n ( τ ))-separating sequen ce (see S ect ion 3 ). Moreov er, if the long range dep endence condition ∆( u n ( τ )) holds for eac h τ > 0, then θ and π d o n ot dep end on τ . The natural app roac h to d o infer en ce on θ and π is to iden tify the clus- ters of exceedances ab o v e a high thresh ol d, th en to ev aluate for eac h clus te r the c haracteristic of in terest and to constru ct estimates fr om these v alues. The tw o common metho ds that are u sed to deﬁn e clus te rs are th e blo c ks and runs d eclustering schemes. The blo c ks declustering sc heme consists in c ho os- ing a blo c k length r n and partitioning the n observ ations into k n = ⌊ n/r n ⌋ blo c ks, where ⌊ x ⌋ d enote s the integ er part of x . Eac h blo c k that cont ains an exceedance is treated as one cluster. The runs declustering scheme con- sists in c ho osing a run length p n , and stipulating th at an y pair of extreme observ ations separated by fewer than p n nonextreme observ ations b elong to the same cluster. The blo c k length r n and the ru n length p n are termed the cluster id entiﬁcatio n scheme sequences an d pla y a key role in determining the asymptotic pr op erties of the estimators. The problem of inf erence on the extremal index has receiv ed a lot of atten tion in the literature. T he ﬁrst blo c ks and ru ns estimators were con- structed b y using diﬀeren t probabilistic c haracterizations of the extremal index (see [ 13 ], Section 8.1, [ 1 , 39 ]). They are d et ermined by t w o s equ ences: the sequ en ce of the thresholds u n ( τ ) and the cluster iden tiﬁcation s c h eme sequence. Their ma jor dr a wb ac k is their dep enden ce on the thr eshold w hic h is b ased on the unkn o w n stationary distribution. E stimat ing this thresh- old is intricat e sin ce, by deﬁnition, it is exceeded by v ery f ew observ ations CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 3 [ 12 ]. T o circumv ent this iss ue, lo w er th r esholds ha v e to b e considered. The follo wing c haracterizat ions (see [ 27 , 31 ]) θ = lim n →∞ s n P  max 1 ≤ i ≤ r n X i > u s n ( τ )  . ( r n τ ) , and θ = lim n →∞ P  max 2 ≤ i ≤ p n X i ≤ u s n ( τ )    X 1 > u s n ( τ )  , where s n = o ( n ), r n = o ( s n ) and p n = o ( s n ), ha v e motiv ated other blo c ks and ru ns estimators [ 21 , 22 , 43 ]; the th r eshold u s n ( τ ) can b e estimate d b y X n −⌊ nτ /s n ⌋ : n . Note that th e estimators are d ete rmin ed by t w o s equ ences as w ell: r n (or p n ) and s n . More recently , new metho ds for identifying clusters of extreme v alues hav e b een introdu ced in [ 26 ] and new estimators of the extremal in dex w h ic h are less s ensitiv e to cluster id en tiﬁcati on scheme se- quences ha v e b een d eriv ed. Ho wev er, to exploit these metho ds, it is necessary to know wh ether the p rocess exhibits either an autoregressiv e or vola til- it y driven dep endence structure and to choose an additional thresh old to iden tify the clusters. In order to eliminate th e cluster ident iﬁcation scheme sequences, [ 16 ] (see also [ 15 ]) prop oses estimators wh ic h are based on th e sequence of the thresholds u r n ( τ ) and on in ter-exceedance times: a least- squares estimator, a maximum-lik elihoo d estimator and a moment estima- tor. It is established that the last-men tioned estimator is weakly consistent for m -dep endent stationary sequences. There are very f ew pap ers which inv estiga te the in ference f or the limiting cluster size pr obabiliti es. In [ 21 ], condition ( 1.1 ) is used to motiv ate the follo wing blo c ks estimators ˆ π n, 1 ( m ; r n , u s n ( τ )) = P k n j =1 1 { Y n,j ( u s n ( τ ))= m } P k n j =1 1 { Y n,j ( u s n ( τ )) > 0 } , (1.2) where Y n,j ( u s n ( τ )) = P j r n i =( j − 1) r n +1 1 { X i >u s n ( τ ) } , s n = o ( n ) and r n = o ( s n ). Let E π ( T ) = P ∞ m =1 T ( m ) π ( m ), where T is a fu nction supp orted on { 1 , 2 , . . . } . The weak consistency of the estimators r n X m =1 T ( m ) ˆ π n, 1 ( m ; r n , X n −⌊ nτ /s n ⌋ : n ) of E π ( T ) is established. Note that they are determined by tw o sequences: r n and s n . I n [ 23 ] the follo wing quan tities are considered ˆ π n, 2 ( m ; r n , u s n ( τ )) = P k n j =1 ( R ( m ) j − R ( m +1) j )1 { Y n,j ( u s n ( τ )) > 0 } P k n j =1 1 { Y n,j ( u s n ( τ )) > 0 } , 4 C. Y. ROBER T where R ( m ) j = ¯ F ( M (1) j ) / ¯ F ( M ( m ) j ) and M ( m ) j is the m th largest v alue of X i , i = ( j − 1) r n + 1 , . . . , j r n . A p artia l comparison with ˆ π n, 1 ( m ; r n , u s n ( τ )) is made und er th e assump tio n that F is kn o w n. Recen tly a n ew metho d has b een prop osed in [ 15 ]: a recursiv e algorithm forms estimates of the limiting cluster size prob ab ilities fr om empirical moments which are based on the join t distributions of the inter-e xceedance times separated b y other in ter- exceedance times. Th ese estimators are only determined by selecting the sequence of thresholds u r n ( τ ). A consistency result for m -dep enden t sta- tionary s equ ences is give n. In the pr esen t pap er we int ro duce new b loc ks estimators of the limiting cluster size pr obabilit ies. The appr oac h is the follo wing. First we estimate the comp ound p robabilities of th e limiting p oin t pro cess. Second we use a declustering (decomp ounding) algorithm to form estimates of th e limiting cluster size probabilities. This idea has b een prop osed r ec entl y in [ 5 ] and [ 6 ] where it is assumed that a sample of th e comp ound P oisson distribution is observ ed (whic h is un f ortunately n ot the case here). More s peciﬁcally , let us den ot e by N ( τ ) E the weak limit of N ( τ ) n ( E ) as n → ∞ when it exists and by p ( τ ) = ( p ( τ ) ( m )) m ≥ 0 its distribu tio n. Let ( ζ i ) i ≥ 1 b e a sequ en ce of p ositiv e i.i.d. in teger-v alued r.v.s with distrib u tion π and η ( θ τ ) b e a r.v. w ith P oisson distr ibution and parameter θ τ s u c h that η ( θ τ ) is ind epend en t of the ( ζ i ) i ≥ 1 . W e ha ve N ( τ ) E d = P η ( θ τ ) i =1 ζ i , with the con v ent ion that the sum equals 0 if the u pp er in dex is smaller th an the lo w er in d ex. The d istribution of N ( τ ) E is giv en by p ( τ ) (0) = P ( η ( θ τ ) = 0) = e − θ τ , (1.3) p ( τ ) ( m ) = m X j =1 P ( η ( θτ ) = j ) P j X i =1 ζ i = m ! = m X j =1 e − θ τ ( θ τ ) j j ! π ∗ j ( m ) , (1.4) m ≥ 1, wher e π ∗ j is the j th conv olution of π , that is, π ∗ j ( m ) =    0 , m < j , X i 1 + ··· + i j = m π ( i 1 ) · · · π ( i j ) , m ≥ j . In r isk theory the aggrega te claim amount is often assu med to ha ve a com- p ound P oisson distrib u tion. Pa njer’s algorithm [ 32 ] is a metho d to compu te recursiv ely th e aggregate clai ms distribution when the distribution of a sin- gle claim is discrete and the distribution of the num b er of claims is P oisson, Binomial or Negativ e-Binomial. F or the limiting comp ound Poi sson distri- bution ( 1.3 )–( 1.4 ), the recur s io n is giv en b y p ( τ ) (0) = e − θ τ , CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 5 p ( τ ) ( m ) = − ln( p ( τ ) (0)) m m X j =1 j π ( j ) p ( τ ) ( m − j ) , m ≥ 1 . Note that the p ( τ ) ( m ) can b e expressed as a function of the π ( j ) , j = 1 , . . . , m . It is p ossib le to rev erse the alg orithm and to ev aluate recursiv ely the π ( m ) from the p ( τ ) ( j ), j = 0 , . . . , m , and the π ( j ), j = 0 , . . . , m − 1, in th e follo wing w a y π ( m ) = − ( p ( τ ) ( m ) + m − 1 ln( p ( τ ) (0)) P m − 1 j =1 j π ( j ) p ( τ ) ( m − j )) ln( p ( τ ) (0)) p ( τ ) (0) , (1.5) m ≥ 1 . Hence, the inv ersion of Panjer’s algorithm pro vides an app ealing r ecur siv e metho d to estimat e the limiting cluster size p robabilitie s. The conte nt of the pap er is organized as follo ws. In Section 2 w e explain ho w we construct the estimators of the limiting cluster size pr ob ab ilities. W e also deriv e estimators of the extremal index. W e emph asiz e th at all our estimators are determined b y one sequence an d one (or t wo ) parameter(s). In Section 3 we p r esen t and discu s s tec hnical conditions whic h are required f or establishing the asymp totic p roper ties. In S ection 4 we giv e results on weak con v ergence of the estimators. In Section 5 we inv estigate the ﬁnite sample b eha vior of the estimators on simulate d data and w e make a comparison with existing estimators. Pro ofs are gathered in a last section. 2. D eﬁning the estimators. In the remainder of the p aper we assu me that u n ( τ ) = F ← (1 − τ /n ). The p resen t approac h to estimating the limit- ing cluster size distribution is b ase d on th e blo c ks d eclustering sc heme. W e divide { 1 , . . . , n } in to k n blo c ks of length r n , I j = { ( j − 1) r n + 1 , . . . , j r n } for j = 1 , . . . , k n , an d a last blo ck I k n +1 = { r n k n + 1 , . . . , n } . The n um b er of observ ations ab o v e the threshold u r n ( τ ) w ithin the j th blo c k is denoted by N ( τ ) r n ,j = X i ∈ I j 1 { X i >u r n ( τ ) } , j = 1 , . . . , k n . Since lim n →∞ E ( N ( τ ) r n ,j ) = τ , the parameter τ can b e in terpreted as the asym p - totic mean num b er of observ atio ns w hic h exceed the lev el u r n ( τ ) for eac h blo c k. Th e empirical d istribution, p ( τ ) n , of the num b er of exceedances within a b loc k is giv en by p ( τ ) n ( m ) = 1 k n k n X j =1 1 { N ( τ ) r n ,j = m } , m ≥ 0 . As men tioned in the introd uction, the main issue when using th ese qu anti- ties f or estimating p ( τ ) is that the th reshold u r n ( τ ) is b ase d on the un kno wn 6 C. Y. ROBER T stationary distribution. It h as to b e estimated from the d ata . W e deﬁne the estimator of p ( τ ) b y ˆ p ( τ ) n ( m ) = 1 k n k n X j =1 1 { ˆ N ( τ ) r n ,j = m } , m ≥ 0 , where ˆ N ( τ ) r n ,j = P i ∈ I j 1 { X i > ˆ u r n ( τ ) } and ˆ u r n ( τ ) = X k n r n −⌊ k n τ ⌋ : k n r n . Let us no w consider the estimators of the limiting cluster size p robabili- ties. T o en sure th at the en tries in ( 1.5 ) are nonnegativ e and that their sum do es not excee d 1 , w e deﬁne r ec ur siv ely ˆ π ( τ ) n ( m ) = max 0 , min χ ( τ ) n ( m ) , 1 − m − 1 X j =1 ˆ π ( τ ) n ( j ) !! , m ≥ 1 , where χ ( τ ) n ( m ) = − ( ˆ p ( τ ) n ( m ) + m − 1 ln( ˆ p ( τ ) n (0)) P m − 1 j =1 j ˆ π ( τ ) n ( j ) ˆ p ( τ ) n ( m − j )) ln( ˆ p ( τ ) n (0)) ˆ p ( τ ) n (0) . W e also deﬁne smo othed versions by b ¯ π n ( m ) = 1 φ − σ Z φ σ ˆ π ( τ ) n ( m ) dτ , m ≥ 1 , for giv en 0 < σ < φ (see [ 35 ] for a similar a ve raging tec hniqu e u sed to re- duce th e asymptotic v ariance of the moment estimator of th e extreme v alue parameter). Finally , let u s deriv e estimators of the extremal index. Th is parameter app ears in diﬀeren t moments of the distr ibutions of N ( τ ) E and ζ 1 (when they exist) P ( N ( τ ) E = 0) = e − θ τ , E ( ζ 1 ) = θ − 1 , V ( N ( τ ) E ) = θ τ E ( ζ 1 ) 2 . Fix an in teger m ≥ 1. W e consider t w o appro ximations of θ θ 2 ( m ) = 1 P m j =1 j π ( j ) and θ ( τ ) 3 ( m ) = P m j =0 ( j − τ ) 2 p ( j ) τ P m j =1 j 2 π ( j ) . Estimators of θ , θ 2 ( m ) and θ ( τ ) 3 ( m ) can b e constructed by equ at ing theoret- ical m omen ts to their empirical coun terparts ˆ θ ( τ ) 1 ,n = − ln( ˆ p ( τ ) n (0)) τ , ˆ θ ( τ ) 2 ,n ( m ) = 1 P m j =1 j ˆ π ( τ ) n ( j ) , ˆ θ ( τ ) 3 ,n ( m ) = P m j =0 ( j − τ ) 2 ˆ p ( τ ) n ( j ) τ P m j =1 j 2 ˆ π ( τ ) n ( j ) . CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 7 ˆ θ ( τ ) 1 ,n can b e seen as a sligh t mo diﬁcation of the estimator in equation (1.5) in [ 39 ]. ˆ θ ( τ ) 2 ,n ( m ) has b een studied in [ 21 ] with ( 1.2 ) as an estimator of the limiting cluster size distribu tion and m = r n . T o the b est of our kn o w ledge , ˆ θ ( τ ) 3 ,n ( m ) seems to b e new. Finally , let us d eﬁne b ¯ θ 1 ,n b y the smo othed v ersion of th e ﬁ rst estimator b ¯ θ 1 ,n = 1 φ − σ Z φ σ ˆ θ ( τ ) 1 ,n dτ . All estimators (resp. smo othed v ersions of the estimators) introd u ced in this section are determin ed by the sequence r n and the p arameter τ (resp. φ and σ ). They provide an interesting alternativ e to the estimators introd uced in [ 16 ] and [ 15 ] wh er e it is only n eeded to select the sequence of the thresh ol ds u r n ( τ ). Note that b oth metho ds share th e same p arsimon y since in our case u r n ( τ ) is estimated b y X k n r n −⌊ k n τ ⌋ : k n r n . 3. T ec hnical conditions. In this section we pr esen t and discus s tec hnical conditions w hic h are r equired for establishing the asymptotic prop erties of the estimators. W e b egin by giving deﬁnitions wh ich are essen tially due to [ 20 , 27 , 33 ]. The stationary sequence ( X n ) is said to ha v e extremal index θ ≥ 0 if, for eac h τ > 0, lim n →∞ P ( N ( τ ) n = 0) = exp( − θ τ ) . Fix an inte ger r ≥ 1 and τ 1 > · · · > τ r > 0. Deﬁne F ( τ 1 ,...,τ r ) p,q as the σ - algebra generated by the ev en ts { X i > u n ( τ j ) } , p ≤ i ≤ q and 1 ≤ j ≤ r , and write α n,l ( τ 1 , . . . , τ r ) ≡ sup {| P ( A ∩ B ) − P ( A ) P ( B ) | : A ∈ F ( τ 1 ,...,τ r ) 1 ,t , B ∈ F ( τ 1 ,...,τ r ) t + l,n , 1 ≤ t ≤ n − l } . The condition ∆( { u n ( τ j ) } 1 ≤ j ≤ r ) is said to hold if lim n →∞ α n,l n ( τ 1 , . . . , τ r ) = 0 for some sequence l n = o ( n ). The long range d epen d ence c ondition ∆( { u n ( τ j ) } 1 ≤ j ≤ r ) implies that extreme even ts situated f ar apart are almost indep endent. Of course, it is imp lie d b y str on g mixing. Supp ose that ∆( { u n ( τ j ) } 1 ≤ j ≤ r ) holds. A sequence of p ositiv e integ ers ( q n ) is said to b e ∆( { u n ( τ j ) } 1 ≤ j ≤ r )-separating if q n = o ( n ) and there exists a sequ ence ( l n ) suc h that l n = o ( q n ) and lim n →∞ nq − 1 n α n,l n ( τ 1 , . . . , τ r ) = 0. W e no w present the tec hnical conditions. The ﬁrst one w ill b e considered for “wea k consistency” of the estimators. Condition ( C0). The stationary sequ en ce ( X n ) has extremal ind ex θ > 0 . ∆( u n ( τ )) holds for eac h τ > 0 and there exists a probabilit y measur e π = ( π ( i )) i ≥ 1 , s uc h that, for i ≥ 1, π ( i ) = lim n →∞ P ( N ( τ ) n ( B n ) = i | N ( τ ) n ( B n ) > 0) , (C0.a) 8 C. Y. ROBER T with B n = (0 , q n /n ], for some ∆( u n ( τ ))-separating sequence ( q n ). Moreov er, there exists a constan t ρ > 2 suc h that, for eac h τ > 0, sup n ≥ 1 E ( N ( τ ) n ( E )) ρ < ∞ . (C0.b) Condition (C0) en s ures that th e exceedance p oin t pro cess N ( τ ) n con v erges in d istribution for every c hoice of τ > 0 (see [ 24 ], Theorem 4.2). Let 0 < v < ρ . C ondition ( C0.b ) implies that ( N ( τ ) n ( E )) v are uniform ly in tegrable and lim n →∞ E ( N ( τ ) n ( E )) v = E ( N ( τ ) E ) v < ∞ . In particular, the ﬁrst an d second momen ts of N ( τ ) E exist (see [ 4 ], p ag e 338). They are giv en b y E ( N ( τ ) E ) = τ and V ( N ( τ ) E ) = θ τ E ( ζ 1 ) 2 . The follo wing set of cond itions will b e considered for c haracterizing the distributional asymptotics of the estimators. Condition (C1). Con d itio n (C0) holds. ∆( u n ( τ 1 ) , u n ( τ 2 )) holds for eac h τ 1 > τ 2 > 0 and there exists a probabilit y m ea sur e π 2 = ( π ( τ 2 /τ 1 ) 2 ( i, j )) i ≥ j ≥ 0 ,i ≥ 1 , suc h that, for i ≥ j ≥ 0 , i ≥ 1, π ( τ 2 /τ 1 ) 2 ( i, j ) = lim n →∞ P ( N ( τ 1 ) n ( B n ) = i, N ( τ 2 ) n ( B n ) = j | N ( τ 1 ) n ( B n ) > 0) , (C1.a) with B n = (0 , q n /n ], for some ∆( u n ( τ 1 ) , u n ( τ 2 ))-separating sequence ( q n ). Let us int ro duce the tw o-lev el exceedance p oin t p rocess N ( τ 1 ,τ 2 ) n = ( N ( τ 1 ) n , N ( τ 2 ) n ) for τ 1 > τ 2 > 0. Condition (C1) en sures that N ( τ 1 ,τ 2 ) n con v erges in distribution to a p oin t pro cess with Laplace transf orm E exp − 2 X i =1 Z E f i dN ( τ i ) ! = exp  − τ 1 θ Z 1 0 (1 − L ( f 1 ( t ) , f 2 ( t ))) dt  , where N ( τ i ) is the i th marginal of the limiting p oin t pro cess, f i ≥ 0 and L is the Laplace transform of π ( τ 2 /τ 1 ) 2 (see Th eo rem 2.5 in [ 33 ] and its pro of ). Let us denote by ( N ( τ 1 ) E , N ( τ 2 ) E ) the w eak limit of ( N ( τ 1 ) n ( E ) , N ( τ 2 ) n ( E )). By considering constan t fun ctio ns f i , we deduce th at ( N ( τ 1 ) E , N ( τ 2 ) E ) d = η ( θ τ 1 ) X i =1 ζ ( τ 2 /τ 1 ) 1 ,i , η ( θ τ 1 ) X i =1 ζ ( τ 2 /τ 1 ) 2 ,i ! , where ( ζ ( τ 2 /τ 1 ) 1 ,i , ζ ( τ 2 /τ 1 ) 2 ,i ) i ≥ 1 is a sequ en ce of i.i.d. inte ger vec tor r.v.s with dis- tribution π ( τ 2 /τ 1 ) 2 and η ( θτ 1 ) is a r.v. with P oisson distribution and parameter CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 9 θ τ 1 suc h that η ( θτ 1 ) is in dep enden t of the ( ζ ( τ 2 /τ 1 ) 1 ,i , ζ ( τ 2 /τ 1 ) 2 ,i ) (see also Theo- rem 2 in [ 29 ]). T he distribu tion p ( τ 1 ,τ 2 ) 2 = ( p ( τ 1 ,τ 2 ) 2 ( i, j )) i ≥ j ≥ 0 of ( N ( τ 1 ) E , N ( τ 2 ) E ) is giv en by p ( τ 1 ,τ 2 ) 2 (0 , 0) = P ( η ( θ τ 1 ) = 0) = e − θ τ 1 , p ( τ 1 ,τ 2 ) 2 ( i, j ) = i X k =1 P ( η ( θτ 1 ) = k ) P k X l =1 ζ ( τ 2 /τ 1 ) 1 ,l = i, k X l =1 ζ ( τ 2 /τ 1 ) 2 ,l = j ! = e − θ τ 1 i X k =1 ( θ τ 1 ) k k ! π ( τ 2 /τ 1 ) , ∗ k 2 ( i, j ) , i ≥ j ≥ 0 , i ≥ 1 , where π ( τ 2 /τ 1 ) , ∗ k 2 is the k th con v olution of π ( τ 2 /τ 1 ) 2 , that is, π ( τ 2 /τ 1 ) , ∗ k 2 ( i, j ) =            0 , i < k , X i 1 + ··· + i k = i j 1 + ··· + j k = j i q ≥ j q ≥ 0 ,i q ≥ 1 , 1 ≤ q ≤ k π ( τ 2 /τ 1 ) 2 ( i 1 , j 1 ) · · · π ( τ 2 /τ 1 ) 2 ( i k , j k ) , i ≥ k . Condition ( C0.b ) imp lie s that C o v ( N ( τ 1 ) E , N ( τ 2 ) E ) = θ τ 1 E ( ζ ( τ 2 /τ 1 ) 1 , 1 ζ ( τ 2 /τ 1 ) 2 , 1 ) is ﬁnite. Condition (C2). Let r > 2 and φ > 0. There exists a constant D = D ( r , φ ) su c h that, for φ ≥ τ 1 ≥ τ 2 ≥ 0, sup n ≥ 1 E ( N ( τ 1 ) n ( E ) − N ( τ 2 ) n ( E )) r ≤ D ( τ 1 − τ 2 ) . (C2.a) Let θ d ≥ 3 r / ( r − (2 + µ )), where 0 < µ < (( r − 2) ∧ 1 / 2). There exists a constan t C > 0 suc h that, for ev ery c hoice of τ 1 > · · · > τ m > 0 , m ≥ 1 , 1 ≤ l ≤ n , α n,l ( τ 1 , . . . , τ m ) ≤ α l := C l − θ d . (C2.b) ( r n ) is sequ ence such that r n → ∞ and r n = o ( n ) and there exists a sequence ( l n ) s atisfying l n = o ( r 2 /r n ) and lim n →∞ nr − 1 n α l n = 0 . (C2.c) Note that condition ( C2.a ) pro vides an inequalit y which is q u ite natur al to pr ov e tightness criteria. Cond itio n ( C2.b ) is satisﬁed by strong-mixing stationary sequences where the mixing co eﬃcie nts v anish at least w ith a h yp erb olic r ate . The und er lyin g id ea to establish th e asymptotic prop erties 10 C. Y. ROBER T of th e estimators is to split the blo c k I j in to a sm all blo c k of length l n and a b ig blo c k of length r n − l n . Cond itio n ( C2.c ) ensures th at l n is su ﬃcie ntly large such that blo c ks that are not adjacent are asymptotically ind ep en den t, but do es not gro w to o fast su c h that the con tribu tio ns of the small blo c ks are n eg ligible. Finally , w e need a condition on the conv ergence rate of r n to inﬁ n it y to guaran tee that the extreme v alue approximati ons are suﬃcien tly accurate. Condition (C3). Let m b e an in teger. T h e s equ ence ( r n ) s atisﬁes lim n →∞ p k n ( τ − r n ¯ F ( u r n ( τ ))) = 0 and lim n →∞ p k n m X l =1 | P ( N ( τ ) r n ( E ) = l ) − p ( τ ) ( l ) | = 0 lo ca lly uniform ly f or τ > 0. Note that, if F is cont inuous, then r n ¯ F ( u r n ( τ )) = τ and the ﬁr st p art of Condition (C3) is obviously satisﬁed. W e no w discuss the example of the ﬁrst order sto c hastic equ at ions with r andom co eﬃcien ts. A sp ecial case is the squared AR CH(1) pro cess introdu ced in [ 14 ]. This pro cess is probably one of th e most prominent ﬁ nancial time s eries mo del of the last t w o decades. Example 3.1. Let X 0 b e a r .v. and let ( A n , B n ), n ≥ 1, b e i.i.d. (0 , ∞ ) 2 - v alued rand om v ectors indep endent of X 0 . Deﬁne X n b y means of the sto c h astic d iﬀerence equation X n = A n X n − 1 + B n , n ≥ 1 . (3.1) F or sak e of simp lic it y , we assume that the distrib ution of ( A 1 , B 1 ) is abso- lutely cont inuous. K este n [ 25 ] prov ed that there exists a r.v. X , in dep enden t of ( A 1 , B 1 ), suc h that X d = A 1 X + B . Assu m e that X 0 has the same distri- bution as X , so that ( X n ) is a strictly stationary sequence. According to Corollary 2.4.1 in [ 8 ], ( X n ) is also strongly mixing and absolutely r eg ular with geometric rates. F urther, supp ose that th ere exist κ > 0 and ξ > 0 suc h th at E A κ 1 = 1 , E ( A κ 1 max(log( A 1 ) , 0)) < ∞ , E A κ + ξ 1 < ∞ and E B κ + ξ 1 ∈ (0 , ∞ ) . Under these momen t assumptions, results of Goldie [ 17 ] sh o w th at th er e exit c > 0 and ρ > 0 suc h th at ¯ F ( x ) = cx − κ (1 + O ( x − ρ )) , as x → ∞ . (3.2) CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 11 W e deduce that u n ( τ ) = ( cn/τ ) 1 /κ (1 + O ( n − ρ/κ )) as n → ∞ . The one-lev el p oin t pro cess of exceedances was studied in [ 19 ] and the multi-le ve l p oin t pro cess of exceedances in [ 33 ]. No w we successive ly verify that our tec hnical conditions hold. Let R ( x ) = ♯ { j ≥ 1 : ˜ X Q j i =1 A i > x } , wh ere P ( ˜ X > x ) = x − κ , x ≥ 1 , and deﬁne θ k = P ( R (1) = k ), k ≥ 0 . Using results in [ 19 ] and in [ 33 ], we see that ∆( u n ( τ )) holds for eac h τ > 0 and that θ = θ 0 and π ( k ) = ( θ k − 1 − θ k ) /θ 0 , k ≥ 1, for any ( q n ) ∆( u n ( τ ))-separating sequ en ce such that q n = n ς with 0 < ς < 1. More- o v er, b y Lemma 6.1 with τ 1 = τ an d τ 2 = 0, we deduce that E ( N ( τ ) n ( E )) 3 < ∞ and that Condition (C0) holds with ρ = 3. By [ 33 ], ∆( u n ( τ 1 ) , u n ( τ 2 )) holds for eac h τ 1 > τ 2 > 0 and θ π ( τ 2 /τ 1 ) 2 ( i, j ) =  P  R (1) = i − 1 , R  τ 1 τ 2  1 /κ  = j  − P  R (1) = i, R  τ 1 τ 2  1 /κ  = j  + τ 2 τ 1  P  R  τ 2 τ 1  1 /κ  = i − 1 , R (1) = j − 1  − P  R  τ 2 τ 1  1 /κ  = i − 1 , R (1) = j  for an y ( q n ) ∆ ( u n ( τ 1 ) , u n ( τ 2 ))-separating sequence s uc h that q n = n ς with 0 < ς < 1. T h erefore, C ondition (C1) h olds . By Lemma 6.1 , E ( N ( τ 1 ) n ( E ) − N ( τ 2 ) n ( E )) 3 ≤ K ( τ 1 − τ 2 ) for φ ≥ τ 1 ≥ τ 2 ≥ 0. There exists a constan t C satisfying ( C2.b ) for any θ d ≥ 9 / (1 − µ ), wh ere 0 < µ < 1 / 2 b ecause ( X n ) is a geomet rically strong-mixing sequence. More- o v er, if r n = n ς with 0 < ς < 1 and l n = n γ with 0 < γ < 2 ς / 3 , then ( C2.c ) is satisﬁed. Therefore, Condition (C2) holds . Under the assumptions on ( A 1 , B 1 ), F is absolutely con tin uous and r n ¯ F ( u r n ( τ )) = τ . Let us us e Lemma 6.2 with q n = ⌊ n α ⌋ , m n = ⌊ n β ⌋ , δ n = ⌊ n γ ⌋ , x n = ⌊ n δ ⌋ and r n = ⌊ n ς ⌋ with 0 < β < α < 1 , 0 < γ < κ − 1 , δ > 0 and 0 < ς < 1, then there exists a constant K suc h that p k n m X l =1 | P ( N ( τ ) r n ( E ) = l ) − p ( τ ) ( l ) | ≤ K n (1 − ζ ) / 2 ( n − χζ + n (1 − α ) ζ η 2 n β ζ / 3 + ϕ n αζ n δζ ξ ) lo ca lly uniformly for τ > 0, with χ = ( α − β ) ∧ (1 − α ) ∧ α ∧ γ ∧ δ ( κ − ǫ ) ∧ ρ/κ , 0 < η < 1, 0 < ϕ < 1 and 0 < ǫ < κ . Finally , c ho ose 1 / (1 + 2 χ ) < ζ < 1 su c h that C on d itio n (C3) h olds . 12 C. Y. ROBER T 4. A symptotic prop erties of the estimators. T o charac terize the asymp - totic prop erties of the estimators, it is con ve nient to introd uce D m σ ,φ ≡ D ([ σ , φ ] , R m ) [resp. D m ≡ D ((0 , ∞ ) , R m )], the s pace of fu nctions from [ σ, φ ] [resp. (0 , ∞ )] to R m whic h are c` agl` a d (left-con tin uous with righ t-limits) equipp ed with the str ong J 1 -top olo gy (see [ 44 ] wh ere the spaces of c` adl` ag fu nctio ns (righ t-con tin uous with left-limits) are equiv alently considered ). L et us recall that w eak con v ergence (whic h will b e denoted b y ⇒ ) in D m is equiv alen t to w eak con v ergence of the restrictions of the sto c h astic pro cesses to any compact [ σ , φ ], 0 < σ < φ < ∞ . W e start this section by giving a “w eak consistency” r esult. Pr opo s ition 4.1. Supp ose that (C0) holds. L et ( r n ) b e a se quenc e such that r n → ∞ and r n = o ( n ) , and 0 < σ < φ < ∞ . Then ( ˆ p ( · ) n (0) , . . . , ˆ p ( · ) n ( m )) ⇒ ( p ( · ) (0) , . . . , p ( · ) ( m )) in D m +1 σ ,φ , ( ˆ π ( · ) n (1) , . . . , ˆ π ( · ) n ( m )) ⇒ ( π (1) , . . . , π ( m )) in D m σ ,φ , ( ˆ θ ( · ) 1 ,n , ˆ θ ( · ) 2 ,n ( m ) , ˆ θ ( · ) 3 ,n ( m )) ⇒ ( θ, θ 2 ( m ) , θ ( · ) 3 ( m )) in D 3 σ ,φ , b ¯ π n ( m ) P → π ( m ) , m ≥ 1 and b ¯ θ 1 ,n P → θ . W e conti nue with a series of r esults leading to a c haracterization of the distributional asymptotics of the estimators of the limiting cluster size p rob- abilities. W e ﬁrst introd uce th e follo wing cen tered pro cesses: e j,n ( · ) = p k n ( p ( · ) n ( j ) − P ( N ( · ) r n , 1 = j )) , j ≥ 0 , ¯ e n ( · ) = p k n ( ¯ p ( · ) n − r n P ( X 1 > u r n ( · ))) , where ¯ p ( τ ) n = ∞ X i =1 ip ( τ ) n ( i ) = 1 k n k n X j =1 N ( τ ) r n ,j = 1 k n r n k n X i =1 1 { X i >u r n ( τ ) } . ¯ p ( · ) n is call ed the tail empirical distribution and ¯ e n ( · ) the tail empirical pro- cess. Th ey are ve ry usefu l to ols for studyin g the asymptotic prop erties of tail index estimators (see, e.g., [ 9 , 34 ]) or for inf erence of multiv ariate extreme v alue distrib u tions [ 18 ]. The weak con ve rgence of the tail emp irical pro cess of strong-mixing (resp . abs olute regular) stationary sequences has b een studied CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 13 b y [ 37 ] (resp. by [ 37 ], [ 10 ] and [ 11 ]). Note that th e absolute regularit y con- dition implies th e strong-mixing condition wh ic h implies ∆( { u n ( τ j ) } 1 ≤ j ≤ r ) for ev ery c hoice of τ 1 > · · · > τ r > 0, r ≥ 1. The follo wing theorem deals with the w eak con ve rgence of the pr ocess E m,n ( · ) = ( e 0 ,n ( · ) , . . . , e m,n ( · ) , ¯ e n ( · )) in D m +2 . It will b e u s eful throughout th is section. Theorem 4.1. Supp ose that (C1) and (C2) hold. Ther e exists a p ath- wise c ontinuous c enter e d Gaussian pr o c e ss E m ( · ) = ( e 0 ( · ) , . . . , e m ( · ) , ¯ e ( · )) with c ovarianc e functions deﬁne d for 0 < τ 2 ≤ τ 1 by: • if i = 0 , . . . , m , co v ( e i ( τ 1 ) , e i ( τ 2 )) = p ( τ 1 ,τ 2 ) 2 ( i, i ) − p ( τ 1 ) ( i ) p ( τ 2 ) ( i ) , co v ( e i ( τ 1 ) , ¯ e ( τ 2 )) = i X j =0 j p ( τ 1 ,τ 2 ) 2 ( i, j ) − τ 2 p ( τ 1 ) ( i ) , co v (¯ e ( τ 1 ) , e i ( τ 2 )) = ∞ X j = i j p ( τ 1 ,τ 2 ) 2 ( j, i ) − τ 1 p ( τ 2 ) ( i ) , • if 0 ≤ i < j ≤ m , co v( e i ( τ 1 ) , e j ( τ 2 )) = − p ( τ 1 ) ( i ) p ( τ 2 ) ( j ) , • if 0 ≤ j < i ≤ m , co v ( e i ( τ 1 ) , e j ( τ 2 )) = p ( τ 1 ,τ 2 ) 2 ( i, j ) − p ( τ 1 ) ( i ) p ( τ 2 ) ( i ) , • and co v( ¯ e ( τ 1 ) , ¯ e ( τ 2 )) = − ln( p ( τ 1 ) (0)) X 0 ≤ j ≤ i, 1 ≤ i ij π ( τ 2 /τ 1 ) 2 ( i, j ) , such that E m,n ⇒ E m in D m +2 . Let us compare the cond iti ons in [ 37 ] that are needed for conv er gence of ¯ e ( · ) in the case of strong-mixing sequences w ith our conditions. First w e ha v e to imp ose that th e threshold, u n , in [ 37 ], is such that u n = O ( u r n ( τ )). Then Condition C1 in [ 37 ] is equiv alent to our condition ( C2.a ). Condition D2 in [ 37 ] is sligh tly we ak er than our condition ( C2.b ) and condition ( C2.c ) since w e also assume th at l n = o ( r 2 /r n ). Condition C3 in [ 37 ] is imp lied b y our Condition (C1) , but it app ears as a natural suﬃcient cond itio n when u n = O ( u r n ( τ )). 14 C. Y. ROBER T No w let u s consid er the estimators of the comp ound pr ob ab ilities and in tro duce the follo wing pro cesses: ˆ e j,n ( · ) = p k n ( ˆ p ( · ) n ( j ) − p ( · ) ( j )) , j ≥ 0. Theorem 4.2. Supp ose that (C1) , (C2) and (C3) hold. L et 0 < σ < φ < ∞ . Then ( ˆ e 0 ,n ( · ) , . . . , ˆ e m,n ( · )) ⇒ ( ˆ e 0 ( · ) , . . . , ˆ e m ( · )) in D m +1 σ ,φ , wher e ˆ e j ( · ) = e j ( · ) − h j ( · ) ¯ e ( · ) and h j ( τ ) = ∂ p ( ¯ τ ) ( j ) /∂ ¯ τ | ¯ τ = τ . Note th at the h j ( · ) satisfy the recursion h 0 ( · ) = p ( · ) (0) ln p (1) (0) , h j ( · ) = − ln( p ( · ) (0)) j j X i =1 iπ ( i )  ln( p (1) (0)) ln( p ( · ) (0)) p ( · ) ( j − i ) + h j − i ( · )  , j ≥ 1 . In ord er to add ress the asymptotic prop erties of the estimators of the limiting cluster size probabilities, w e construct sev eral pr o cesses. F ollo wing [ 6 ], we deﬁne recursively the pro cesses ˆ d j ( · ) using the intermediate p rocesses w j ( · ) := p ( · ) ( j ) (ln( p ( · ) (0)) p ( · ) (0)) 2 ˆ e 0 ( · ) − 1 j p ( · ) (0) j − 1 X i =0 ( j − i ) π ( j − i )ˆ e i ( · ) − 1 ln( p ( · ) (0)) p ( · ) (0) ˆ e j ( · ) − 1 j p ( · ) (0) j − 1 X i =1 ip ( · ) ( j − i ) ˆ d j ( · ) b y ˆ d 0 ( · ) = − ˆ e 0 ( · ) /p ( · ) (0) and for j ≥ 1, ˆ d j ( · ) :=                                    w j ( · ) , if π ( j ) > 0 and j X i =1 π ( i ) < 1, min ( w j ( · ) , − j − 1 X i =1 ˆ d i ( · ) ) , if π ( j ) > 0 and j X i =1 π ( i ) = 1, max { 0 , w j ( · ) } , if π ( j ) = 0 and j X i =1 π ( i ) < 1, max ( 0 , min ( w j ( · ) , − j − 1 X i =1 ˆ d i ( · ) )) , if π ( j ) = 0 and j X i =1 π ( i ) = 1. CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 15 Note that the pro cess ˆ d j ( · ) dep ends on the s upp ort of the limiting cluster s ize distribution. I t is not in general a Gaussian p rocess b ecause of th e trunca- tions in its construction, except if π ( i ) > 0 for i = 1 , . . . , j and P j i =1 π ( i ) < 1. In the follo wing corollary we deriv e the wea k conv er gence of the pro cesses ˆ d j,n ( · ) = p k n ( ˆ π ( · ) n ( j ) − π ( j )) , j ≥ 1, and th e asymptotic b ehavio r of ¯ d j,n = p k n ( b ¯ π n ( j ) − π ( j )) , j ≥ 1. Corollar y 4.1. Supp ose that (C1) , (C2) and (C3) hold. L et 0 < σ < φ < ∞ . Then ( ˆ d 1 ,n ( · ) , . . . , ˆ d m,n ( · )) ⇒ ( ˆ d 1 ( · ) , . . . , ˆ d m ( · )) in D m σ ,φ and ( ¯ d 1 ,n , . . . , ¯ d m,n ) d →  1 φ − σ Z φ σ ˆ d 1 ( τ ) dτ , . . . , 1 φ − σ Z φ σ ˆ d m ( τ ) dτ  . W e end this section by fo cusing on the estimators of the extremal index. Corollar y 4.2. Supp ose that (C1) , (C2) and (C3) hold. L et 0 < σ < φ < ∞ . Then p k n ( ˆ θ ( · ) 1 ,n − θ ) ⇒ − 1 ( · ) p ( · ) (0) ˆ e 0 ( · ) in D 1 σ ,φ , p k n ( ˆ θ ( · ) 2 ,n ( m ) − θ 2 ( m )) ⇒ − ( θ 2 ( m )) 2 m X j =1 j ˆ d j ( · ) in D 1 σ ,φ , p k n ( ˆ θ ( · ) 3 ,n ( m ) − θ ( · ) 3 ( m )) ⇒ ( P m j =0 ( j − ( · )) 2 ˆ e j ( · ) − θ ( · ) 3 ( m ) P m j =1 j 2 ˆ d j ( · )) ( · ) P m j =1 j 2 π ( j ) in D 1 σ ,φ and p k n ( b ¯ θ 1 ,n − θ ) d → − 1 φ − σ Z φ σ 1 τ p ( τ ) (0) ˆ e 0 ( τ ) dτ . 16 C. Y. ROBER T Note th at the asymptotic v ariance of ˆ θ ( τ ) 1 ,n is giv en by τ − 2 e θ τ − 2 θ τ − 1 + θ 3 τ ∞ X j =1 j 2 π ( j ) ! . It can b e estimated by using the estimators of the limiting clus te r probabil- ities ˆ π ( τ ) n ( j ) and the estimator of the extremal index ˆ θ ( τ ) 1 ,n . 5. S im ulation study . A s imulation study is conducted to inv estiga te the p erformance of the estimators on large samples and to make a comparison with existing estimators. (i) Performanc e on lar ge samples. Data are simulate d from thr ee station- ary Mark o v pro cesses: • a squared ARCH(1) pr ocess: X n = ( η + λX n − 1 ) Z 2 n , n ≥ 2 , where Z n are i.i.d. standard Gaussian r.v.s, η = 2 × 10 − 5 , λ = 0 . 5 and X 1 is a r .v. dra wn from the stationary d istr ibution of the c hain. The limiting cluster size probabilities and the extremal index h av e b een computed b y s im ulations in [ 19 ]: π (1) = 0 . 751, π (2) = 0 . 168, π (3) = 0 . 055, π (4) = 0 . 014, π (5) = 0 . 008, θ = 0 . 727. • a max-AR (1) p rocess: X n = m ax { (1 − θ ) X n − 1 , W n } , n ≥ 2, where W n are i.i.d. unit F r´ echet r .v.s, θ = 0 . 5 and X 1 = W 1 /θ . By [ 33 ], π (1) = 0 . 5, π (2) = 0 . 25, π (3) = 0 . 125, π (4) = 0 . 0625, π (5) = 0 . 031, θ = 0 . 5. • an AR(1) pro cess with uniform marginal: X n = r − 1 X n − 1 + ε n , n ≥ 2, where ( ε n ) are i.i.d. r.v.s uniformly d istributed on { 0 , 1 /r, . . . , ( r − 1) /r } , r = 4 and X 1 is uniformly distributed on (0 , 1). By [ 33 ], π (1) = 0 . 75, π (2) = 0 . 1875, π (3) = 0 . 04 69, π (4) = 0 . 0117, π (5) = 0 . 0029, θ = 0 . 75. T o compare the p erformance of the estimators, 500 sequences of length n = 2000 were simulat ed from the three pro cesses. W e h a ve considered the ratios ˆ π (1) n ( j ) /π ( j ) for j = 1 , . . . , 5, ˆ θ (1) 1 ,n /θ , and ˆ θ (1) j,n ( m ) /θ for j = 2 , 3 and m = 8. The graph s sho w the a ve rage o ver the 500 samples. In Figures 1 and 2 the means and the ro ot mean squared errors (RMSE) of the ratios are p lot ted as a fun ctio n of k n . Th e bias of ˆ π (1) n (1) is sm all and appro ximativ ely stable with resp ect to k n for the three pro cesses. T he biases of ˆ π (1) n (2) and ˆ π (1) n (3) are small for th e squared AR CH(1) pro cess and the max-AR(1) pro cess bu t large for the AR(1) pro cess. F or j ≥ 4, the biases of the estimators can b e relativ ely large and it seems v ery diﬃcult to hav e go o d estimates of π ( j ) in the case of a d ata set of length 2000. The RMSE of the ratios increase dramatically with j b ecause of the biases. Note also that a minimum of the RMSE with resp ect to k n can not alw a ys b e found. An optimal choi ce of k n based on th e RMSE criterion will d epen d on the p rocess and on the limiting cluster size p robabilities. CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 17 Fig. 1. Me ans of the r atios of the cluster size pr ob abilities ˆ π (1) n (1) /π (1) , ˆ π (1) n (2) /π (2) , ˆ π (1) n (3) /π (3) , ˆ π (1) n (4) /π (4) , ˆ π (1) n (5) /π (5) , and me ans of the r atios of the extr emal index ˆ θ (1) 1 ,n /θ , ˆ θ (1) 2 ,n (8) /θ and ˆ θ (1) 3 ,n (8) /θ as a function of k n = 50 , . . . , 250 for the squar e d ARCH(1) pr o c ess (—–), the max- AR(1) pr o c ess (- - - -) and the AR(1) pr o c ess ( · · · · ). The gr aphs show the aver age over 500 samples of length n = 2000 . The bias of ˆ θ (1) 1 ,n is lo w er than those of ˆ θ (1) 2 ,n ( m ) and ˆ θ (1) 3 ,n ( m ) for the squ ared AR CH(1) pr o cess and th e max-AR(1) pro cess. But for the AR(1) pro cess, the bias of ˆ θ (1) 3 ,n ( m ) is the smallest. ˆ θ (1) 1 ,n and ˆ θ (1) 2 ,n ( m ) p erform in the same wa y in terms of RMS E and b etter than ˆ θ (1) 3 ,n ( m ). (ii) Comp arison with existing estimators on lar ge samples. Data are sim- ulated from the squared ARCH(1) pro cess deﬁned b elo w. 500 sequences of length n = 2000 w ere also used. F or the limiting cluster p r obabilitie s com- 18 C. Y. ROBER T Fig. 2. RMSE of the r atios of the cluster size pr ob abilities ˆ π (1) n (1) /π (1) , ˆ π (1) n (2) /π (2) , ˆ π (1) n (3) /π (3) , ˆ π (1) n (4) /π (4) , ˆ π (1) n (5) /π (5) , and RM SE of the r atios of the extr emal index ˆ θ (1) 1 ,n /θ , ˆ θ (1) 2 ,n (8) /θ and ˆ θ (1) 3 ,n (8) /θ as a function of k n = 50 , . . . , 250 for the squar e d ARCH(1) pr o c ess (—–), the max- AR(1) pr o c ess (- - - -) and the AR(1) pr o c ess ( · · · · ). The gr aphs show the aver age over 500 samples of length n = 2000 . parisons are made b et wee n ˙ π n ( i ) = ˆ π (1) n ( j ), b ¯ π n ( j ) with σ = 0 . 7 and φ = 1 . 3, Hsing’s estimators ˆ π n, 1 ( j ) with n/s n = k n / 2 and F err o’s estimators ˜ π n ( j ) with N = k n (see [ 15 ], equation (4.12)). F or the extremal in d ex comparisons are made b et w een ˙ θ n = ˆ θ (1) 1 ,n , b ¯ θ 1 ,n with σ = 0 . 7 and φ = 1 . 3, F err o and Segers’ estimator ˜ θ n ( u ) with u = X n − k n +1: n (see [ 16 ], equation (5)), Hsing’s estimator ˜ θ n with n/s n = k n / 2 (see [ 21 ], page 137) and the r u ns estima- CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 19 Fig. 3. RMSE of the r atios of the cluster size pr ob abilities ˙ π n (1) /π (1) , ˙ π n (2) /π (2) , ˙ π n (3) /π (3) , ˙ π n (4) /π (4) , ˙ π n (5) /π (5) as a f unct ion of k n = 50 , . . . , 250 , for ˙ π n = ˆ π (1) n (- - - -), ˙ π n = b ¯ π n (——), ˙ π n = ˜ π n (F err o’ s estimators · · · · ) and ˙ π n = ˆ π n, 1 (Hsing’s estimator – – –). RMSE of the r atios of the extr emal index ˙ θ n /θ as a function of k n = 50 , . . . , 250 , for ˙ θ n = ˆ θ (1) 1 ,n (- - - -), ˙ θ n = b ¯ θ 1 ,n (——), ˙ θ n = ˜ θ n (F err o and Se gers’ estimator · · · · ), ˙ θ n = ˜ θ n (Hsing’s estimator – – – ) and ˙ θ n = ˆ θ R n (runs estimator - – - –). tor ˆ θ R n ( p, u ) with p = ⌊ r n / 6 ⌋ , u = X n −⌊ n/s n ⌋ : n and n/s n = k n / 2 (see [ 43 ], page 282). In Figure 3 the RMSE of the r ati os ˙ π n ( i ) /π ( i ) for i = 1 , . . . , 5, and ˙ θ n /θ are plotted. F or the limiting cluster s ize p robabilities and the extremal in- dex, the s m oothed ve rsions b ¯ π n and b ¯ θ 1 ,n p erform uniformly b etter than the unsmo othed estimators ˆ π (1) n and ˆ θ (1) 1 ,n whic h p erform un iformly b etter than 20 C. Y. ROBER T the other estimators [except for π (2), where Hsing’s estimator should b e preferred to th e unsmo othed estimator]. As F erro and Segers’ estimators, our estimators only r equire the c hoice of a s equence, but their p erformance is more fa v orable. 6. Pro ofs. Th r oughout we let K b e a generic constan t whose v alue ma y c hange from line to line. Lemma 6.1. Consider the ﬁrst or der sto chastic e quation with r andom c o eﬃcients of Example 3.1 . Ther e exists a c onstant K such that, for φ ≥ τ 1 ≥ τ 2 ≥ 0 and n ≥ 1 , E ( N ( τ 1 ) n ( E ) − N ( τ 2 ) n ( E )) 3 ≤ K ( τ 1 − τ 2 ) . Pr oof . Let I n ( τ 1 , τ 2 ) = ( u n ( τ 1 ) , u n ( τ 2 )]. By u s ing the same arguments as in th e pro of of Lemma 4.1 in [ 10 ], w e can s h o w that there exists a constan t K such that, for φ ≥ τ 1 ≥ τ 2 ≥ 0 and n ≥ 1, c i,j = P ( X j ∈ I n ( τ 1 , τ 2 ) | X i ∈ I n ( τ 1 , τ 2 )) ≤ K  1 n + ϕ j − i  , j ≥ i ≥ 1 , where ϕ = E A ξ 1 < 1 for ξ ∈ (0 , κ ). By the stationary and Mark o v pr operty , w e get E ( N ( τ 1 ) n ( E ) − N ( τ 2 ) n ( E )) 3 ≤ 3! n X i,j ≥ 1 ,i + j ≤ n +1 E 1 { X 1 ∈ I n ( τ 1 ,τ 2 ) } 1 { X i ∈ I n ( τ 1 ,τ 2 ) } 1 { X i + j − 1 ∈ I n ( τ 1 ,τ 2 ) } ≤ 3!( τ 1 − τ 2 ) X i,j ≥ 1 ,i + j ≤ n +1 c 1 ,i c i,i + j − 1 ≤ 3!( τ 1 − τ 2 ) K 2 X i,j ≥ 1 ,i + j ≤ n +1  1 n + ϕ j − 1  1 n + ϕ i − 1  ≤ K ( τ 1 − τ 2 ) .  Lemma 6.2. Consider the ﬁrst or der sto chastic e quation with r andom c o eﬃcients of Example 3.1 . L et ( q n ) , ( m n ) , ( δ n ) , ( x n ) b e se quenc es of in- te gers such that q n → ∞ and q n = o ( n ) , m n → ∞ and m n = o ( q n ) , δ n → ∞ and nδ − κ n → ∞ and x n → ∞ as n → ∞ . Then for e ach l ≥ 0 , ther e exists a c onstant K such that | P ( N ( τ ) n ( E ) = l ) − p ( τ ) ( l ) | ≤ K  m n q n + q n n + 1 q n + n q n η 2 m n / 3 + q ⌊ nδ − κ n ⌋ nδ − κ n + δ − 1 n + ϕ q n x ξ n + x − ( κ − ǫ ) n + n − ρ/κ  CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 21 lo c al ly u niform ly f or τ > 0 , wher e 0 < η < 1 , 0 < ϕ < 1 , 0 < ξ < κ and 0 < ǫ < κ . Pr oof . W rite d ( n, l ) = | P ( N ( τ ) n ( E ) = l ) − p ( τ ) ( l ) | . Let θ ( τ ) n = n × P ( N ( τ ) n ( B n ) > 0) / ( τ q n ), w here B n = (0; q n /n ]. Let ζ ( τ ) i,n , i ≥ 1, b e i.i.d. in teger- v alued r.v.s suc h that P ( ζ ( τ ) i,n = m ) = P ( N ( τ ) n ( B n ) = m | N ( τ ) n ( B n ) > 0) , m ≥ 1 , and η ( θ ( τ ) n τ ) b e a Poisson r.v. with parameter θ ( τ ) n τ and indep end ent of the ζ ( τ ) i,n . W e hav e that d ( n, l ) ≤      P ( N ( τ ) n ( E ) = l ) − P η ( θ ( τ ) n τ ) X i =1 ζ ( τ ) i,n = l !      +      P η ( θ ( τ ) n τ ) X i =1 ζ ( τ ) i,n = l ! − p ( τ ) ( l )      =: I l + I I l . By u sing Theorem 2 in [ 30 ], we dedu ce that I l ≤ 2 τ m n q n + 2 τ q n n + n q n min { 6 α 2 / 3 n,m n ( τ ); β n,m n ( τ ) } , where β n,l ( τ ) ≡ su p 1 ≤ t ≤ n − l E su p | P ( B |F ( τ ) 1 ,t ) − P ( B ) : B ∈ F ( τ ) t + l,n | . Note that, sin ce ( X n ) is a geometrically absolute regular sequ ence, th ere exists a constan t 0 < η < 1 such that, for ev ery c hoice of τ > 0 and 1 ≤ l ≤ n , α n,l ( τ ) ≤ β n,l ( τ ) ≤ O ( η l ). By ( 1.3 ) and ( 1.4 ), we dedu ce that there exist constants K 1 ,l > 0 and K 2 ,l > 0 suc h th at, lo ca lly unif orm ly for τ > 0, I I l ≤ K 1 ,l | θ ( τ ) n − θ 0 | + K 2 ,l l X k =1 | π ( τ ) n ( k ) − π ( k ) | ≤ K 1 ,l | θ ( τ ) n − θ 0 | + 2 θ − 1 0 K 2 ,l l +1 X k =1      θ 0 q n X j = k π ( τ ) n ( j ) − θ k − 1      . Let θ ( τ ) k ,n = P ( N ( τ ) n ( B n ) = k | X 0 > u n ( τ )). Note that | θ ( τ ) n − θ 0 | ≤ | θ ( τ ) n − θ ( τ ) 0 ,n | + | θ ( τ ) 0 ,n − θ 0 | =: I I a + I I b 22 C. Y. ROBER T and f or k ≥ 1,      θ 0 q n X j = k π ( τ ) n ( j ) − θ k − 1      ≤     θ 0 P ( N ( τ ) n ( B n ) ≥ k ) P ( N ( τ ) n ( B n ) ≥ 1) − θ 0 θ ( τ ) n θ ( τ ) k − 1 ,n     +     θ 0 θ ( τ ) n − 1     θ ( τ ) k − 1 ,n + | θ ( τ ) k − 1 ,n − θ k − 1 | =: I I c k + I I d k + I I e k . By using the same argum ents as f or the pro of of L emm a 2.4 in [ 33 ], we ha ve for k ≥ 1 | P ( N ( τ ) n ( B n ) ≥ k ) − ( q n − k + 1) P ( N ( τ ) n ( B n ) = k − 1 , X 0 > u n ( τ )) | ≤ k P ( M 0 ,q n > u n ( τ ) , M q n , 2 q n > u n ( τ )) , where M i,j = max { X l : l = i + 1 , . . . , j } . It follo ws that for k ≥ 1 I I c k ≤ k θ 0 θ ( τ ) n nP ( M 0 ,q n > u n ( τ ) , M q n , 2 q n > u n ( τ )) τ q n + k − 1 q n θ 0 θ ( τ ) n and I I a ≤ nP ( M 0 ,q n > u n ( τ ) , M q n , 2 q n > u n ( τ )) / ( τ q n ) . No w observ e that P ( M 0 ,q n > u n ( τ ) , M q n , 2 q n > u n ( τ )) = P ( {{ M 0 ,q n − m n > u n ( τ ) } ∪ { M q n − m n ,q n > u n ( τ ) }} ∩ { M q n , 2 q n > u n ( τ ) } ) ≤ P ( M q n − m n ,q n > u n ( τ )) + α n,m n ( τ ) + P 2 ( M 0 ,q n > u n ( τ )) ≤ τ m n n + α n,m n ( τ ) +  τ q n n  2 θ 2 0 and, therefore, I I a + l +1 X k =1 (I I c k + I I d k ) ≤ K  m n q n + n q n α n,m n ( τ ) + q n n + 1 q n  . Let σ k = P ∞ j = k θ j = P ( R (1) ≥ k ) = R ∞ 1 P ( ♯ { j ≥ 1 : Q j i =1 A i > x − 1 } ≥ k ) κ × x − κ − 1 dx . W e ha v e that θ k − 1 = σ k − σ k − 1 for k ≥ 1. Then I I b ≤     q n X j =0 θ ( τ ) j,n − σ 0      +      q n X j =1 θ ( τ ) j,n − σ 1      I I e k ≤      q n X j = k − 1 θ ( τ ) j,n − σ k − 1      +      q n X j = k θ ( τ ) j,n − σ k     , k ≥ 1 . Let u s deﬁne the probabilit y measure, Q n , on (1 , ∞ ) b y Q n ( dx ) = P (( u n ( τ )) − 1 X 0 ∈ dx ) /P (( u n ( τ )) − 1 X 0 > 1) . CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 23 As in [ 19 ], w e in tro duce the pr ocess (∆ n ) deﬁn ed by ∆ 0 = 0 and ∆ n = A n ∆ n − 1 + B n , n ≥ 1. W e h a ve that ∆ n ≥ 0 and X n = X 0 Q n i =1 A i + ∆ n for n ≥ 1 . Let B k ( q n , (∆ j ) j =1 ,...,q n , Q n ) = Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i + ∆ j > u n ( τ ) ) ≥ k | ( u n ( τ )) − 1 X 0 = x ! Q n ( dx ) . Note th at P ( N ( τ ) n ( B n ) ≥ k | X 0 > u n ( τ )) = B k ( q n , (∆ j ) j =1 ,...,q n , Q n ) and that      q n X j = k θ ( τ ) j,n − σ k      ≤ | B k ( q n , (∆ j ) j =1 ,...,q n , Q n ) − B k ( q n , (0) j =1 ,...,q n , Q n ) | + | B k ( q n , (0) j =1 ,...,q n , Q n ) − B k ( ∞ , (0) j =1 ,..., ∞ , Q n ) | + | B k ( ∞ , (0) j =1 ,..., ∞ , Q n ) − B k ( ∞ , (0) j =1 ,... , ∞ , Q ) | . W e no w consid er successive ly eac h term of the u p p er b ound: (i) On the one hand, we ha ve that Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i + ∆ j > u n ( τ ) ) ≥ k      ( u n ( τ )) − 1 X 0 = x ! × Q n ( dx ) ≥ Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i > u n ( τ ) ) ≥ k      ( u n ( τ )) − 1 X 0 = x ! × Q n ( dx ) . On the other h and, w e h a ve th at ( ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i + ∆ j > u n ( τ ) ) ≥ k ) ⊂ ( ♯ ( 1 ≤ j ≤ q n : (( X 0 j Y i =1 A i > u n ( τ )(1 − δ − 1 n ) ) ∪ { ∆ j > δ − 1 n u n ( τ ) } ) ≥ k ) ⊂ ( ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i > u n ( τ )(1 − δ − 1 n ) ) 24 C. Y. ROBER T + ♯ { 1 ≤ j ≤ q n : ∆ j > δ − 1 n u n ( τ ) } ≥ k ) ⊂ q n [ l =0 { ♯ { 1 ≤ j ≤ q n : ∆ j > δ − 1 n u n ( τ ) } = l } ∩ { ♯ { 1 ≤ j ≤ q n : ∆ j > δ − 1 n u n ( τ ) } ≥ ( k − l ) ∨ 0 } . Then Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i + ∆ j > u n ( τ ) ) ≥ k      ( u n ( τ )) − 1 X 0 = x ! × Q n ( dx ) ≤ Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i > u n ( τ )(1 − δ − 1 n ) ) ≥ k      ( u n ( τ )) − 1 X 0 = x ! Q n ( dx ) + Z ∞ 1 P ( ♯ { 1 ≤ j ≤ q n : { ∆ j > δ − 1 n u n ( τ ) }} > 0) Q n ( dx ) . Note th at ∆ j ≤ X j for j ≥ 1 and, therefore, Z ∞ 1 P ( ♯ { 1 ≤ j ≤ q n : { ∆ j > δ n u n ( τ ) }} > 0) Q n ( dx ) ≤ P ( M q n > δ − 1 n u n ( τ )) ≤ K q ⌊ nδ − κ n ⌋ nδ − κ n if q n → ∞ and nδ − κ n → ∞ as n → ∞ . Moreo v er, by a change of v ariable, we ha v e that Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : X 0 j Y i =1 A i > u n ( τ )(1 − δ − 1 n ) ) ≥ k      ( u n ( τ )) − 1 X 0 = x ! × Q n ( dx ) = (1 + o (1)) (1 − δ − 1 n ) κ Z ∞ (1 − δ − 1 n ) − 1 P ♯ ( 1 ≤ j ≤ q n : j Y i =1 A i > x − 1 ) ≥ k ! Q n ( dx ) . Since the density f u nction of Q n is un iformly b oun ded in a neigh b orho o d of 1, w e d educe that R (1 − δ − 1 n ) − 1 1 Q n ( dx ) ≤ K δ − 1 n and it follo ws that | B k ( q n , (∆ j ) j =1 ,...,q n , Q n ) − B k ( q n , (0) j =1 ,...,q n , Q n ) | ≤ K  q ⌊ nδ − κ n ⌋ nδ − κ n + δ − 1 n  . CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 25 (ii) Let ϕ = E A ξ 1 < 1 for ξ ∈ (0 , κ ). W e ha v e that Z ∞ 1 P ♯ ( j ≥ 1 : j Y i =1 A i > x − 1 ) ≥ k ! Q n ( dx ) = Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : j Y i =1 A i > x − 1 ) + ( j > q n : j Y i =1 A i > x − 1 ) ≥ k ! × Q n ( dx ) ≤ Z ∞ 1 P ♯ ( 1 ≤ j ≤ q n : j Y i =1 A i > x − 1 ) ≥ k ! Q n ( dx ) + Z ∞ 1 P ♯ ( j > q n : j Y i =1 A i > x − 1 ) > 0 ! Q n ( dx ) . It follo ws that | B k ( q n , (0) j =1 ,...,q n , Q n ) − B k ( ∞ , (0) j =1 ,..., ∞ , Q n ) | ≤ Z ∞ 1 P ♯ ( j > q n : j Y i =1 A i > x − 1 ) > 0 ! Q n ( dx ) ≤ Z x n 1 P ∞ _ j = q n +1 j Y i =1 A i > x − 1 ! Q n ( dx ) + Q n ( x n , ∞ ) ≤ ∞ X j = q n +1 ϕ j x ξ n + Q n ( x n , ∞ ) ≤ ϕ q n x ξ n 1 − ϕ + K x κ − ǫ n b y Chebyshev’s inequalit y and P otter’s b ounds. (iii) Let f k ( x ) = P ( ♯ {{ j ≥ 1 : Q j i =1 A i > x − 1 } ≥ k ). Since the distrib ution of A 1 is absolutely con tin uous, f k is diﬀerentiable. Then w e hav e B k ( ∞ , (0) j =1 ,..., ∞ , Q n ) − B k ( ∞ , (0) j =1 ,..., ∞ , Q ) = Z ∞ 1 f k ( x )( Q n ( dx ) − Q ( dx )) = Z ∞ 1 f (1) k ( x )( Q n ( x, ∞ ) − Q ( x, ∞ )) dx, where f (1) k is the ﬁrs t deriv ative of f k . But, by equation ( 3.2 ), sup x ≥ 1 | ( Q n ( x, ∞ ) × Q − 1 ( x, ∞ ) − 1) | ≤ K n − ρ/κ and we dedu ce that | B k ( ∞ , (0) j =1 ,..., ∞ , Q n ) − B k ( ∞ , (0) j =1 ,..., ∞ , Q ) | ≤ K n − ρ/κ . Putting the inequalities toge ther yields I I b + l +1 X k =1 I I e k ≤ K  q ⌊ nδ − κ n ⌋ nδ − κ n + δ − 1 n + ϕ q n x ξ n + x − ( κ − ǫ ) n + n − ρ/κ  26 C. Y. ROBER T and th e result follo ws.  W e no w d eﬁne the big blo c ks I △ j and th e small blo cks I ∗ j b y I △ j = { ( j − 1) r n + 1 , . . . , j r n − l n } , I ∗ j = { j r n − l n + 1 , . . . , j r n } , j = 1 , . . . , k n . Let us in tro duce the follo wing r.v.s asso ciate d with th e big and small blo c ks: N ( τ ) , △ r n ,j = X i ∈ I △ j 1 { X i >u r n ( τ ) } , N ( τ ) , ∗ r n ,j = X i ∈ I ∗ j 1 { X i >u r n ( τ ) } , j = 1 , . . . , k n , p ( τ ) , △ n ( i ) = 1 k n k n X j =1 1 { N ( τ ) , △ r n ,j = i } , p ( τ ) , ∗ n ( i ) = 1 k n k n X j =1 (1 { N ( τ ) , △ r n ,j = i − N ( τ ) , ∗ r n ,j ,N ( τ ) , ∗ r n ,j > 0 } − 1 { N ( τ ) , △ r n ,j = i,N ( τ ) , ∗ r n ,j > 0 } ) , ¯ p ( τ ) , △ n = 1 k n k n X j =1 N ( τ ) , △ r n ,j , ¯ p ( τ ) , ∗ n = 1 k n k n X j =1 N ( τ ) , ∗ r n ,j . It is easily seen that p ( τ ) n ( i ) = p ( τ ) , △ n ( i ) + p ( τ ) , ∗ n ( i ) and ¯ p ( τ ) n = ¯ p ( τ ) , △ n + ¯ p ( τ ) , ∗ n . T o pr o ve Prop osition 4.1 , w e will need the three f ol lo wing lemmas. Th e ﬁrst lemma can b e derived from Lemma 1 in [ 7 ]. Lemma 6.3. L et p 1 , p 2 , p 3 b e p ositive numb ers such that p − 1 1 + p − 1 2 + p − 1 3 = 1 . Supp ose that Y and Z ar e r andom variables me asur able with r esp e ct to the σ -algebr a F ( τ 1 ,...,τ r ) 1 ,m , F ( τ 1 ,...,τ r ) m + l,n r esp e ctively ( 1 ≤ m ≤ n − l ) and assume further that k Y k p 1 = ( E | Y | p 1 ) 1 /p 1 < ∞ , k Z k p 2 = ( E | Z | p 2 ) 1 /p 2 < ∞ . Then | Co v ( Y , Z ) | ≤ 10( α n,l ( τ 1 , . . . , τ r )) 1 /p 3 k Y k p 1 k Z k p 2 . Lemma 6.4. Supp ose that (C0) holds. L e t ( r n ) b e a se quenc e su c h that r n → ∞ and r n = o ( n ) . Then p ( τ ) n ( i ) P → p ( τ ) ( i ) and ¯ p ( τ ) n P → τ . CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 27 Pr oof . Since r n → ∞ , ∆( u r n ( τ )) holds and there exists a sequence ( l n ) suc h that l n → ∞ , l n = o ( r n ) and α r n ,l n ( τ ) → 0. Let ε > 0. By Chebyshev’s inequalit y , P ( | p ( τ ) , ∗ n ( i ) | > ε ) ≤ ε − 1 ( P ( N ( τ ) , △ r n , 1 = i − N ( τ ) , ∗ r n , 1 , N ( τ ) , ∗ r n , 1 > 0) + P ( N ( τ ) , △ r n , 1 = i, N ( τ ) , ∗ r n , 1 > 0)) ≤ 2 ε − 1 P ( N ( τ ) , ∗ r n , 1 > 0) ≤ 2 ε − 1 P [ i ∈ I ∗ 1 { X i > u r n ( τ ) } ! ≤ 2 ε − 1 τ l n /r n → 0 and P ( | ¯ p ( τ ) , ∗ n | > ε ) ≤ ε − 1 E ( N ( τ ) , ∗ r n , 1 ) ≤ ε − 1 τ l n /r n → 0 . Hence, p ( τ ) , ∗ n ( i ) P → 0 and ¯ p ( τ ) , ∗ n P → 0. No w let us show that p ( τ ) , △ n ( i ) P → p ( τ ) ( i ) and ¯ p ( τ ) , △ n P → τ . Since lim n →∞ P ( N ( τ ) , ∗ r n , 1 = i ) = 0 and lim n →∞ E ( N ( τ ) , ∗ r n , 1 ) = 0, we deduce b y condition ( C0.b ) th at lim n →∞ P ( N ( τ ) , △ r n , 1 = i ) = p ( τ ) ( i ) and lim n →∞ E ( N ( τ ) , △ r n , 1 ) = τ . Therefore, it suﬃces to sho w that p ( τ ) , △ n ( i ) − P ( N ( τ ) , △ r n , 1 = i ) P → 0 and ¯ p ( τ ) , △ n − E ( N ( τ ) , △ r n , 1 ) P → 0 . W e ha v e P ( | p ( τ ) , △ n ( i ) − P ( N ( τ ) , △ r n , 1 = i ) | > ε ) ≤ ε − 2 E ( p ( τ ) , △ n ( i ) − P ( N ( τ ) , △ r n , 1 = i )) 2 ≤ 2( k n ε ) − 2 X 1 ≤ j ≤ l ≤ k n | Co v (1 { N ( τ ) , △ r n ,l = i } , 1 { N ( τ ) , △ r n ,j = i } ) | . By u sing Lemma 6.3 with p 1 = ∞ , p 2 = ∞ , p 3 = 1 , we get P ( | p ( τ ) , △ n ( i ) − P ( N ( τ ) , △ r n , 1 = i ) | > ε ) ≤ K ( k n ε ) − 2 k n + k n − 1 X j =1 ( k n − j ) α r n ,l n +( j − 1) r n ( τ ) ! k 1 { N ( τ ) , △ r n , 1 = i } k 2 ∞ ≤ K ε − 2 ( k − 1 n + α r n ,l n ( τ )) → 0 . In the same w a y , by using Lemma 6.3 with p 1 = ρ , p 2 = ρ , p 3 = ρ/ ( ρ − 2), w e get P ( | ¯ p ( τ ) , △ n − E ( N ( τ ) , △ r n , 1 ) | > ε ) ≤ ε − 2 E ( ¯ p ( τ ) , △ n − E ( N ( τ ) , △ r n , 1 )) 2 ≤ 2( k n ε ) − 2 X 1 ≤ j ≤ l ≤ k n | Co v ( N ( τ ) , △ r n ,j , N ( τ ) , △ r n ,l ) | 28 C. Y. ROBER T ≤ K ( k n ε ) − 2 k n + k n − 1 X j =1 ( k n − j )( α r n ,l n +( j − 1) r n ( τ )) 1 − 2 ρ − 1 ! k N ( τ ) , △ r n , 1 k 2 ρ ≤ K ε − 2 ( k − 1 n + ( α r n ,l n ( τ )) 1 − 2 ρ − 1 ) k N ( τ ) r n , 1 k 2 ρ . Observe th at su p n ≥ 1 E ( N ( τ ) r n , 1 ) ρ < ∞ by condition ( C0.b ) and 1 − 2 ρ − 1 > 0 to conclude.  Lemma 6.5. Supp ose that (C0) holds. L e t ( r n ) b e a se quenc e su c h that r n → ∞ and r n = o ( n ) . Then ( p ( · ) n (0) , . . . , p ( · ) n ( m ) , ¯ p ( · ) n ) ⇒ ( p ( · ) (0) , . . . , p ( · ) ( m ) , ( · )) in D m +2 . Pr oof . Let u s ﬁ rst recall that conv ergence in D m +2 is equ iv alen t to con v ergence in D m +2 σ ,φ for all choice of p ositiv e σ and φ , 0 < σ < φ < ∞ . Moreo v er, sin ce ( p ( · ) (0) , . . . , p ( · ) ( m ) , ( · )) is a d ete rmin istic elemen t of D m +2 σ ,φ , w e only need to prov e that p ( · ) n ( i ) ⇒ p ( · ) ( i ) in D 1 σ ,φ , i = 0 , . . . , m , and ¯ p ( · ) n ⇒ ( · ) in D 1 σ ,φ . By Theorem 13.1 in [ 3 ], it suﬃces to pro v e that the ﬁnite- dimensional distribu tio ns con v erge and that a tigh tness criterion h olds . It is easily seen that the ﬁ r st condition is satisﬁed by u sing Lemma 6.4 . W e only need to c hec k that the ( p ( · ) n ( i )) n ≥ 1 , i = 0 , . . . , m , and ( ¯ p ( · ) n ) n ≥ 1 are tigh t in D 1 σ ,φ . F ollo wing Section 12 in [ 3 ], we call a set { τ i } a δ -sp arse if it s at isﬁes σ = τ 0 < · · · < τ w = φ and min 1 ≤ i ≤ w ( τ i − τ i − 1 ) ≥ δ , and we deﬁne for q ∈ D 1 σ ,φ w ′ ( q , δ ) = inf { t i } max 1 ≤ i ≤ w sup s,t ∈ ( τ i − 1 ,τ i ] | q ( s ) − q ( t ) | . By using T h eorem 13.2 in [ 3 ] and its coroll ary , p ( · ) n ( i ) is tigh t in D 1 σ ,φ if and only if the t w o follo wing conditions hold: (i) for eac h τ in a set that is dense in [ σ, φ ] and con tains σ , lim a →∞ lim sup n P ( p ( τ ) n ( i ) > a ) = 0 , (ii) for eac h ε > 0, lim δ → 0 lim sup n P ( w ′ ( p ( · ) n ( i ) , δ ) > ε ) = 0. Condition (i) is satisﬁed since p ( τ ) n ( i ) P → p ( τ ) ( i ) < 1 for eac h τ ∈ [ σ , φ ] (b y Lemma 6.4 ). Let us now consider cond itio n (ii). Let δ < φ − σ and d eﬁne M δ = ⌊ ( φ − σ ) δ − 1 ⌋ + 1, τ δ l = σ + l δ for 0 ≤ l < M δ and τ δ M δ = φ . Note that τ 7→ P i j =0 p ( τ ) n ( j ) is a nonincr easing function, and then sup τ ,τ ′ ∈ ( τ δ l − 1 ,τ δ l ]     i X j =0 ( p ( τ ) n ( j ) − p ( τ ′ ) n ( j ))     ≤ i X j =0 ( p ( τ δ l − 1 ) n ( j ) − p ( τ δ l ) n ( j )) . CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 29 It follo ws that w ′ i X j =0 p ( · ) n ( j ) , δ ! ≤ max 1 ≤ l ≤ M δ i X j =0 ( p ( τ δ l − 1 ) n ( j ) − p ( τ δ l ) n ( j )) . If i ≥ 1 , w e ha v e P ( w ′ ( p ( · ) n ( i ) , δ ) > ε ) ≤ P w ′ i X j =0 p ( · ) n ( j ) , δ ! > ε 2 ! + P w ′ i − 1 X j =0 p ( · ) n ( j ) , δ ! > ε 2 ! ≤ P max 1 ≤ l ≤ M δ i X j =0 ( p ( τ δ l − 1 ) n ( j ) − p ( τ δ l ) n ( j )) > ε 2 ! + P max 1 ≤ l ≤ M δ i − 1 X j =0 ( p ( τ δ l − 1 ) n ( j ) − p ( τ δ l ) n ( j )) > ε 2 ! . If i = 0 , w e ha v e P ( w ′ ( p ( · ) n ( i ) , δ ) > ε ) ≤ P  max 1 ≤ l ≤ M δ ( p ( τ δ l − 1 ) n (0) − p ( τ δ l ) n (0)) > ε  . By u sing Lemma 6.4 , we get max 1 ≤ l ≤ M δ i X j =0 ( p ( τ δ l − 1 ) n ( j ) − p ( τ δ l ) n ( j )) P → max 1 ≤ l ≤ M δ i X j =0 ( p ( τ δ l − 1 ) ( j ) − p ( τ δ l ) ( j )) , whic h is less than ε/ 2 for small δ , since τ 7→ P i j =0 h j ( τ ) is a con tin uous and b ounded fu nctio n on [ σ, φ ]. Th us, we deduce th at lim δ → 0 lim sup n P ( w ′ ( p ( · ) n ( i ) , δ ) > ε ) = 0 . Condition (ii) is satisﬁed and p ( · ) n ( i ) is tight in D 1 σ ,φ . No w n ote that τ 7→ ¯ p ( τ ) n is a nondecreasing fun cti on and ∂ ¯ p ( τ ) /∂ τ = 1 . The argumen ts for ¯ p ( · ) n run similarly . W e conclude that ( p ( · ) n (0) , . . . , p ( · ) n ( m ) , ¯ p ( · ) n ) w eakly con v erges in D m +2 σ ,φ , an d then in D m +2 .  Pr oof of Pr opos ition 4.1 . The generalized in ve rse of ¯ p ( · ) n is giv en by ¯ p ( ¯ τ ) , ← n = in f ( τ ≥ 0 : r n k n X i =1 1 { X i >F ← (1 − τ /r n ) } ≥ k n ¯ τ ) = r n ¯ F ( X k n r n −⌊ k n ¯ τ ⌋ : k n r n ) since F ← ( F ( X k n r n −⌊ k n ¯ τ ⌋ : k n r n )) = X k n r n −⌊ k n ¯ τ ⌋ : k n r n . I t is a c` agl` ad f unction on [ σ , φ ]. Note that for ¯ τ ∈ [ σ , φ ] and n such that ⌊ k n ¯ τ ⌋ ≤ k n r n , ˆ p ( ¯ τ ) n ( m ) = p ( ¯ p ( ¯ τ ) , ← n ) n ( m ) , m ≥ 0 . 30 C. Y. ROBER T Let D ↑ ,σ ,φ (resp. D σ ,φ ↑ ,σ ,φ , C ↑ ,σ ,φ , C σ ,φ ↑ ,σ ,φ ) b e th e space of n ondecreasing func- tions fr om [ σ, φ ] to R (resp. nondecreasing f u nctions fr om [ σ, φ ] to [ σ, φ ] , con- tin uous nondecreasing fu nctions from [ σ , φ ] to R , con tin uous nondecreasing functions f rom [ σ, φ ] to [ σ, φ ] ). Let us int ro duce the map Υ from D ↑ ,σ ,φ to D σ ,φ ↑ ,σ ,φ taking h in to max( σ, min( h ← , φ )). It is con tin uous at C σ ,φ ↑ ,σ ,φ . Let us d enote by ¯ p ( · ) , ← n,b the function Υ( ¯ p ( · ) n ). By Lemma 6.5 and the contin uous mapping theorem (CMT ), it follo ws that ¯ p ( · ) , ← n,b ⇒ Υ(( · )) = ( · ) in D σ ,φ ↑ ,σ ,φ . Moreo v er, the comp ositio n map from D m +1 σ ,φ × D σ ,φ ↑ ,σ ,φ to D m +1 σ ,φ taking ( g , h ) in to g ◦ h is con tin uous at ( g , h ) ∈ C m +1 σ ,φ × C σ ,φ ↑ ,σ ,φ (see, e.g., [ 2 ], page 145). It follo ws b y th e CMT th at ( p ( ¯ p ( · ) , ← n,b ) n (0) , . . . , p ( ¯ p ( · ) , ← n,b ) n ( m )) ⇒ ( p ( · ) (0) , . . . , p ( · ) ( m )) in D m +1 σ ,φ . No w w e ha v e sup τ ∈ [ σ,φ ] | p ( ¯ p ( τ ) , ← n ) n ( j ) − p ( ¯ p ( τ ) , ← n,b ) n ( j ) | ≤ sup τ , ¯ τ ∈ [ ¯ p ( σ ) , ← n ,σ ] | p ( τ ) n ( j ) − p ( ¯ τ ) n ( j ) | 1 { ¯ p ( σ ) , ← n <σ } ∨ sup τ , ¯ τ ∈ [ φ, ¯ p ( φ ) , ← n ] | p ( τ ) n ( j ) − p ( ¯ τ ) n ( j ) | 1 { ¯ p ( φ ) , ← n >φ } . Since the w eak limit of ( p ( · ) n ( j )) n ≥ 1 is conti nuous at σ and φ , ¯ p ( σ ) , ← n P → σ and ¯ p ( φ ) , ← n P → φ , w e dedu ce that sup τ ∈ [ σ,φ ] | p ( ¯ p ( ¯ τ ) , ← n ) n ( j ) − p ( ¯ p ( τ ) , ← n,b ) n ( j ) | P → 0 , or, equ iv alen tly , p ( ¯ p ( · ) , ← n ) n ( j ) − p ( ¯ p ( · ) , ← n,b ) n ( j ) ⇒ 0 in D 1 σ ,φ . Fin ally , we get ( ˆ p ( · ) n (0) , . . . , ˆ p ( · ) n ( m )) ⇒ ( p ( · ) (0) , . . . , p ( · ) ( m )) in D m +1 σ ,φ . T o p ro v e wea k con ve rgence of ( ˆ π ( · ) n (1) , . . . , ˆ π ( · ) n ( m )) in D m σ ,φ , we p roceed b y induction. Firs t note that by Lemma 6.5 lim n →∞ P ([ ˆ p ( φ ) n (0) , ˆ p ( σ ) n (0)] ∈ (0 , 1)) = 1. W e d educe b y the CMT that χ ( · ) n (1) = − ˆ p ( · ) n (1) ln( ˆ p ( · ) n (0)) ˆ p ( · ) n (0) ⇒ − p ( · ) (1) ln( p ( · ) (0)) p ( · ) (0) = π (1) CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 31 in D 1 σ ,φ , ˆ π ( · ) n (1) = max(0 , min( χ ( · ) n (1) , 1) ) ⇒ π (1) in D 1 σ ,φ and ( ˆ p ( · ) n (0) , ˆ p ( · ) n (1) , ˆ π ( · ) n (1)) ⇒ ( p ( · ) (0) , p ( · ) (1) , π (1)) in D 3 σ ,φ . No w assume that we hav e already sho wn that ( ˆ p ( · ) n (0) , . . . , ˆ p ( · ) n ( j ) , ˆ π ( · ) n (1) , . . . , ˆ π ( · ) n ( j − 1)) ⇒ ( p ( · ) (0) , . . . , p ( · ) ( j ) , π (1) , . . . , π ( j − 1)) in D 2 j σ ,φ . Let us d eﬁne the maps Ψ j from D 2 j σ ,φ to D 1 σ ,φ taking f ( · ) = ( f i ( · )) i =1 ,..., 2 j in to Ψ j ( f ( · )) = − ( f j +1 ( · ) + j − 1 ln( f 1 ( · )) P j − 1 i =1 if i + j +1 ( · ) f j − i +1 ( · )) ln( f 1 ( · )) f 1 ( · ) . Note th at χ ( · ) n ( j ) = Ψ j ( ˆ p ( · ) n (0) , . . . , ˆ p ( · ) n ( j ) , ˆ π ( · ) n (1) , . . . , ˆ π ( · ) n ( j − 1)) and that Ψ j is con tin uous on the space of contin u ous fun ctio ns from [ σ, φ ] to (0 , 1) × R 2 j − 1 . It follo ws by th e CMT that χ ( · ) n ( j ) ⇒ π ( j ) in D 1 σ ,φ . Let u s recall that ˆ π ( · ) n ( j ) = max 0 , min χ ( · ) n ( j ) , 1 − j − 1 X i =1 ˆ π ( · ) n ( i ) !! . W e conclude by the C MT that ˆ π ( · ) n ( j ) ⇒ π ( j ) in D 1 σ ,φ and ( ˆ p ( · ) n (0) , . . . , ˆ p ( · ) n ( j + 1) , ˆ π ( · ) n (1) , . . . , ˆ π ( · ) n ( j )) ⇒ ( p ( · ) (0) , . . . , p ( · ) ( j + 1) , π (1) , . . . , π ( j )) in D 2( j +1) σ ,φ . The induction is established and ( ˆ π ( · ) n (1) , . . . , ˆ π ( · ) n ( m )) ⇒ ( π (1) , . . . , π ( m )) in D m σ ,φ . Finally , by using aga in the CMT, we deduce that ( ˆ θ ( · ) 1 ,n , ˆ θ ( · ) 2 ,n ( m ) , ˆ θ ( · ) 3 ,n ( m )) ⇒ ( θ, θ 2 ( m ) , θ ( · ) 3 ( m )) in D 3 σ ,φ , b ¯ π n ( m ) P → π ( m ), m ≥ 1, and b ¯ θ 1 ,n P → θ .  32 C. Y. ROBER T Let u s now deﬁn e e △ i,n ( τ ) = p k n ( p ( τ ) , △ n ( i ) − P ( N ( τ ) , △ r n ,j = i )) , e ∗ i,n ( τ ) = p k n (1 { N ( τ ) , △ r n ,j = i − N ( τ ) , ∗ r n ,j ,N ( τ ) , ∗ r n ,j > 0 } − P ( N ( τ ) , △ r n ,j = i − N ( τ ) , ∗ r n ,j , N ( τ ) , ∗ r n ,j > 0)) − p k n (1 { N ( τ ) , △ r n ,j = i,N ( τ ) , ∗ r n ,j > 0 } − P ( N ( τ ) , △ r n ,j = i, N ( τ ) , ∗ r n ,j > 0) ) , ¯ e △ n ( τ ) = p k n ( ¯ p ( τ ) , △ n − ( r n − l n ) P ( X i > u r n ( τ ))) , ¯ e ∗ n ( τ ) = p k n ( ¯ p ( τ ) , ∗ n − l n P ( X i > u r n ( τ ))) , E △ m,n ( τ ) = ( e △ 0 ,n ( τ ) , . . . , e △ m,n ( τ ) , ¯ e △ n ( τ )) , E ∗ m,n ( τ ) = ( e ∗ 0 ,n ( τ ) , . . . , e ∗ m,n ( τ ) , ¯ e ∗ n ( τ )) . W e ha v e e j,n ( · ) = e △ j,n ( · ) + e ∗ j,n ( · ) and ¯ e n ( · ) = ¯ e △ n ( · ) + ¯ e ∗ n ( · ). Th e pro of of Theorem 4.1 is no w presen ted in a series of three lemmas. Lemma 6.6. Supp ose that (C2) holds. L et τ > 0 . Then E ∗ m,n ( τ ) P → 0 . Pr oof . By ( C2.c ), there exists a sequence ( l n ) s atisfying l n = o ( r 2 /r n ) and lim n →∞ nr − 1 n α l n = 0. W e ha v e that E ( e ∗ i,n ( τ )) 2 ≤ 2 k − 1 n E k n X j =1 (1 { N ( τ ) , △ r n ,j = i − N ( τ ) , ∗ r n ,j ,N ( τ ) , ∗ r n ,j > 0 } − P ( N ( τ ) , △ r n ,j = i − N ( τ ) , ∗ r n ,j , N ( τ ) , ∗ r n ,j > 0)) ! 2 + 2 k − 1 n E k n X j =1 (1 { N ( τ ) , △ r n ,j = i,N ( τ ) , ∗ r n ,j > 0 } − P ( N ( τ ) , △ r n ,j = i, N ( τ ) , ∗ r n ,j > 0)) ! 2 =: 2(I 1 + I 2 ) . Let 2 < v < r . By usin g Lemma 6.3 with p 1 = v , p 2 = v , p 3 = v / ( v − 2), we get I 1 ≤ 2 k − 1 n X 1 ≤ j ≤ l ≤ k n Co v (1 { N ( τ ) , △ r n ,j = i − N ( τ ) , ∗ r n ,j ,N ( τ ) , ∗ r n ,j > 0 } , 1 { N ( τ ) , △ r n ,l = i − N ( τ ) , ∗ r n ,l ,N ( τ ) , ∗ r n ,l > 0 } ) ≤ K k − 1 n k n + k n − 1 X j =1 ( k n − j )( α r n , ( j − 1) r n ( τ )) 1 − 2 v − 1 ! CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 33 × k 1 { N ( τ ) , △ r n , 1 = i − N ( τ ) , ∗ r n , 1 ,N ( τ ) , ∗ r n , 1 > 0 } k 2 v ≤ K 1 + k n − 1 X j =1 α 1 − 2 v − 1 ( j − 1) r n ! ( P ( N ( τ ) , ∗ r n , 1 > 0)) 2 /v ≤ K 1 + ∞ X j =0 α 1 − 2 v − 1 j ! ( l n ¯ F ( u r n ( τ ))) 2 /v ≤ K  l n r n  2 /v , since P ∞ j =0 α 1 − 2 v − 1 j < ∞ . S imilarly , I 2 ≤ K ( l n /r n ) 2 /v . T herefore, P ( | e ∗ i,n ( τ ) | > ε ) ≤ ε − 2 E ( e ∗ i,n ( τ )) 2 ≤ K ( l n /r n ) 2 /v → 0 . By u sing Lemma 6.3 with p 1 = v , p 2 = v , p 3 = v / ( v − 2), w e get E ( ¯ e ∗ n ( τ )) 2 ≤ K k n k n + k n − 1 X j =1 ( k n − j )( α r n , ( r n − l n )+( j − 1) r n ( τ )) 1 − 2 v − 1 ! × k N ( τ ) , ∗ r n , 1 − l n ¯ F ( u r n ( τ )) k 2 v ≤ K 1 + k n − 1 X j =1 ( α r n , ( r n − l n )+( j − 1) r n ( τ )) 1 − 2 v − 1 ! × k N ( τ ) , ∗ r n , 1 − l n ¯ F ( u r n ( τ )) k 2 v . By T heorem 4.1 in [ 40 ] [equation (4.4)], we ha ve E | N ( τ ) , ∗ r n , 1 − l n ¯ F ( u r n ( τ )) | v ≤ K l v/ 2 n k 1 { X 1 >u r n ( τ ) } − ¯ F ( u r n ( τ )) k v r ≤ K  l n r 2 /r n  v/ 2 → 0 . Putting the inequalities ab o ve together yields E ∗ m,n ( τ ) P → 0.  Lemma 6.7. Supp ose that (C1) and (C2) hold. L e t r ≥ 1 and τ 1 > · · · > τ r > 0 . Then ( E m,n ( τ 1 ) , . . . , E m,n ( τ r )) d → ( E m ( τ 1 ) , . . . , E m ( τ r )) . Pr oof . Since b y Lemma 6.6 E ∗ m,n ( τ ) P → 0, we only pro v e that ( E △ m,n ( τ 1 ) , . . . , E △ m,n ( τ r )) d → ( E m ( τ 1 ) , . . . , E m ( τ r )) . By applying the Cr amer–W old device, it suﬃces to p ro v e that, for λ h,j ∈ R , h = 1 , . . . , r and i = 0 , . . . , m + 1, r X h =1 m X i =0 λ h,i e △ i,n ( τ h ) + λ h,m +1 ¯ e △ n ( τ h ) ! d → r X h =1 m X i =0 λ h,i e i ( τ h ) + λ h,m +1 ¯ e ( τ h ) ! . 34 C. Y. ROBER T Let f j,n = r X h =1 m X i =0 λ h,i (1 { N ( τ h ) , △ r n ,j = i } − P ( N ( τ h ) , △ r n ,j = i )) + r X h =1 λ h,m +1 ( N ( τ h ) , △ r n ,j − ( r n − l n ) P ( X 1 > u r n ( τ h ))) . By u sing recursiv ely Lemma 6.3 with p 1 = ∞ , p 2 = ∞ , p 3 = 1 , we get      E exp ( − i u √ k n k n X j =1 f j,n ) − k n Y j =1 E exp  − i u √ k n f j,n       ≤ K k n α r n ,l n ( τ 1 , . . . , τ r ), whic h tends to 0 by cond iti on ( C2.c ). Th is implies that the f j,n can b e considered as i.i.d. r.v.s. By condition ( C0.b ) and Minko w ski’s inequalit y , lim n →∞ E | f j,n | ρ < ∞ where ρ > 2. Therefore, P k n j =1 E | f j,n | ρ ( P k n j =1 E ( f j,n ) 2 ) ρ/ 2 = 1 k ρ/ 2 − 1 n E | f 1 ,n | ρ ( E ( f 1 ,n ) 2 ) ρ/ 2 → 0 and Lyap ouno v’s condition h olds (see, e.g., [ 4 ], page 362). It follo ws that ( k n E ( f 1 ,n ) 2 ) − 1 / 2 P k n i =1 f i,n con v erges in d istribution to a stand ard Gaussian random v ariable. By Condition (C1) , ( N ( τ 1 ) n ( E ) , N ( τ 2 ) n ( E )) ⇒ ( N ( τ 1 ) E , N ( τ 2 ) E ) and the limiting second cen tral moments of the r.v.s 1 { N ( τ h ) , △ r n , 1 = i } and N ( τ h ) , △ r n , 1 , h = 1 , . . . , r , exist. S imple calculations yield th e co v ariance functions giv en in Th eo rem 4.1 .  Lemma 6.8. Supp ose that (C1) and (C2) hold. Then ( E m,n ( · )) n ≥ 1 is tight in D m +2 σ ,φ . Pr oof . W e us e similar arguments as for the second p art of th e p roof of Theorem 22.1 in [ 2 ]. The tightness criterion whic h is considered is the follo wing (see Theorem 15.5 and Theorem 8.3 in [ 2 ]): ( E m,n ( · )) n ≥ 1 is tight in D m +2 σ ,φ if: (i) for eac h p ositiv e η , there exists an a suc h that P ( | E m,n ( φ ) | 1 > a ) ≤ η, n ≥ 1 , where | E | 1 = P m +1 j =0 | E j | ; (ii) letting ε > 0 and η > 0, ther e exists δ > 0 and an integer n 0 suc h that P  sup τ 2 ≤ τ 1 ≤ τ 2 + δ | E m,n ( τ 1 ) − E m,n ( τ 2 ) | 1 > ε  ≤ η δ , n ≥ n 0 , for all τ 2 ∈ [ σ, φ ]. CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 35 Moreo v er, by Th eo rem 15.5 in [ 2 ], it follo ws that the wea k limit of a subsequence E m,n ′ ( · ) b elongs a.s. to C m +2 σ ,φ . Condition (i) is satisﬁed since E m,n ( φ ) d → E m ( φ ). Let us consid er cond i- tion (ii). Note that P  sup τ 2 ≤ τ 1 ≤ τ 2 + δ | E m,n ( τ 1 ) − E m,n ( τ 2 ) | 1 > ε  ≤ m X i =0 P  sup τ 2 ≤ τ 1 ≤ τ 2 + δ | e i,n ( τ 1 ) − e i,n ( τ 2 ) | > ε m + 1  + P  sup τ 2 ≤ τ 1 ≤ τ 2 + δ | ¯ e n ( τ 1 ) − ¯ e n ( τ 2 ) | > ε m + 1  ≤ 2 m X i =0 P sup τ 2 ≤ τ 1 ≤ τ 2 + δ      i X j =0 ( e j,n ( τ 1 ) − e j,n ( τ 2 ))      > ε 2( m + 1) ! + P  sup τ 2 ≤ τ 1 ≤ τ 2 + δ | ¯ e n ( τ 1 ) − ¯ e n ( τ 2 ) | > ε m + 1  and it su ﬃ ce s to c hec k the tigh tness criterion for eac h P i j =0 e j,n ( · ), i = 0 , . . . , m and for ¯ e n ( · ). No w we s im p ly indicate the mo diﬁcations to b e m ade in the pro of of Theorem 22.1 in [ 2 ] to establish th at condition (ii) holds. Let 2 < v < p < r ≤ ∞ and ε > 0. Assume that θ d > v / ( v − 2) and θ d ≥ ( p − 1) r/ ( r − p ). (i) Let σ ≤ τ 2 < τ 1 ≤ φ and deﬁne S i ( τ 1 , τ 2 ; k n ) := p k n i X j =0 ( e j,n ( τ 1 ) − e j,n ( τ 2 )) ! . By T heorem 4.1 in [ 40 ] [equation (4.3)], we ha ve that E | S i ( τ 1 , τ 2 ; k n ) | p ≤ K ( k p/ 2 n ( P ( N ( τ 2 ) r n , 1 ≤ i < N ( τ 1 ) r n , 1 )) p/v + k 1+ ε n ( P ( N ( τ 2 ) r n , 1 ≤ i < N ( τ 1 ) r n , 1 )) p/r ) ≤ K ( k p/ 2 n ( P ( N ( τ 1 ) r n , 1 − N ( τ 2 ) r n , 1 > 1) ) p/v + k 1+ ε n ( P ( N ( τ 1 ) r n , 1 − N ( τ 2 ) r n , 1 > 1)) p/r ) ≤ K ( k p/ 2 n ( E ( N ( τ 1 ) r n , 1 − N ( τ 2 ) r n , 1 )) p/v + k 1+ ε n ( E ( N ( τ 1 ) r n , 1 − N ( τ 2 ) r n , 1 )) p/r ) ≤ K ( k p/ 2 n ( τ 1 − τ 2 ) p/v + k 1+ ε n ( τ 1 − τ 2 ) p/r ) . Let η = p / 2 − (1 + ε ). If 0 < ǫ < 1 and ǫ/k η n ≤ ( τ 1 − τ 2 ) p (1 /v − 1 /r ) , we get E      i X j =0 ( e j,n ( τ 1 ) − e j,n ( τ 2 ))      p ! ≤ K ǫ − 1 ( τ 1 − τ 2 ) p/v , 36 C. Y. ROBER T whic h replaces equ ation (22.15 ) of [ 2 ]. (ii) Let ξ j,n := ( N ( τ 1 ) r n ,j − N ( τ 2 ) r n ,j − ( E N ( τ 1 ) r n ,j − E N ( τ 2 ) r n ,j )) and deﬁn e S ( τ 1 , τ 2 ; k n ) := p k n ( ¯ e n ( τ 1 ) − ¯ e n ( τ 2 )) = k n X j =1 ξ j,n . By T heorem 4.1 in [ 40 ] [equation (4.3)], we ha ve that E | S ( τ 1 , τ 2 ; k n ) | p ≤ K ( k p/ 2 n k ξ 1 ,n k p v + k 1+ ε n k ξ 1 ,n k p r ) . No w for v > 2 , | ξ 1 ,n | v ≤ 2 v (( N ( τ 1 ) r n , 1 − N ( τ 2 ) r n , 1 ) v + ( E N ( τ 1 ) r n , 1 − E N ( τ 2 ) r n , 1 ) v ) . F or large n and for σ ≤ τ 2 < τ 1 ≤ φ , | ξ 1 ,n | v ≤ K (( N ( τ 1 ) r n , 1 − N ( τ 2 ) r n , 1 ) v + ( τ 1 − τ 2 )) . By cond iti on ( C2.a ), we get E ( | ξ 1 ,n | λ ) ≤ K ( τ 1 − τ 2 ) for 2 ≤ λ ≤ r and we deduce that E | S ( τ 1 , τ 2 ; k n ) | p ≤ K ( k p/ 2 n ( τ 1 − τ 2 ) p/v + k 1+ ε n ( τ 1 − τ 2 ) p/r ) . Therefore, if ǫ < 1 and ǫ/k η n ≤ ( τ 1 − τ 2 ) p (1 /v − 1 /r ) , we ha v e that E ( | ¯ e n ( τ 1 ) − ¯ e n ( τ 2 ) | p ) ≤ K ǫ − 1 ( τ 1 − τ 2 ) p/v , whic h also rep laces equation (22.15) of [ 2 ]. (iii) W e replace equation (22.17) in [ 2 ] by      i X j =0 ( e j,n ( τ 1 ) − e j,n ( τ 2 ))      ≤      i X j =0 ( e j,n ( τ 2 + δ ) − e j,n ( τ 2 ))      + δ p k n , | ¯ e n ( τ 1 ) − ¯ e n ( τ 2 ) | ≤ | ¯ e n ( τ 2 + δ ) − ¯ e n ( τ 2 ) | + δ p k n , for τ 2 ≤ τ 1 ≤ τ 2 + δ , b y using monoton y argument s as in [ 2 ]. (iv) W e need to replace (22.1 9) of [ 2 ] by  ǫ k η n  r v/ ( p ( r − v )) ≤ p < ǫ √ k n and to assume that η r v p ( r − v ) = r v ( r − v )  1 2 − (1 + ε ) p  > 1 2 . Since θ d has to b e larger than ( p − 1) r / ( r − p ) wh ic h is increasing in p and p > v , we let p = v (1 + ε ) and c ho ose v su ch that r v ( r − v )  1 2 − (1 + ε ) p  = 1 2 (1 + ε ) . CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 37 It follo ws th at v = (3 + ε ) r / ( r + (1 + ε )). Then the inequalities θ d > v / ( v − 2) and θ d ≥ ( p − 1) r/ ( r − p ) b ecome θ d > 3 + ε 1 + ε r r − 2 and θ d ≥ ((2 + ε ) 2 − 2) r − (1 + ε ) r − (2 + ε )(1 + ε ) , whic h are satisﬁed if ε < (( r − 2) ∧ 1 / 2) / 4 and θ d ≥ 3 r r − 2(1 + 2 ε ) . Ev erything else remains the same as for the pro of of Theorem 22.1 in [ 2 ]. Finally , c ho ose µ = 4 ε .  Pr oof o f Theorem 4.1 . W eak con v ergence in D m +2 of a sto c hastic pro cess is equiv alent to we ak con v ergence of the restrictions of the sto c hastic pro cess to an y compact [ σ, φ ] w ith 0 < σ < φ < ∞ in D m +2 σ ,φ . The con ve rgence of the ﬁn ite dimensional d istributions of E m,n ( · ) is established b y Lemma 6.7 and the tigh tness of ( E m,n ( · )) n ≥ 1 in D m +2 σ ,φ b y Lemm a 6.8 . W eak conv ergence in D m +2 σ ,φ follo ws by Theorem 13.1 in [ 3 ]. By Theorem 15.5 in [ 2 ], we deduce that E m ( · ) ∈ C m +2 .  Pr oof o f Theore m 4.2 . Let ˜ e j,n ( · ) := p k n ( p ( · ) n ( j ) − p ( · ) ( j )) = e j,n ( · ) + p k n ( P ( N ( · ) r n , 1 = j ) − p ( · ) ( j )) . Since sup τ ∈ [ σ,φ ] | √ k n ( P ( N ( τ ) r n , 1 = j ) − p ( τ ) ( j )) | → 0 [by Condition (C3) ], we deduce that ( ˜ e 0 ,n ( · ) , . . . , ˜ e m,n ( · )) ⇒ ( e 0 ( · ) , . . . , e m ( · )) in D m +1 σ ,φ . By using the function Υ , the comp ositio n map, the same arguments as in the pro of of Prop osition 4.1 and Theorem 4.1 , w e deduce that ( ˜ e 0 ,n ( ¯ p ( · ) , ← n,b ) , . . . , ˜ e m,n ( ¯ p ( · ) , ← n,b )) ⇒ ( e 0 ( · ) , . . . , e m ( · )) in D m +1 σ ,φ . No w note that sup τ ∈ [ σ,φ ] | ˜ e j,n ( ¯ p ( · ) , ← n ) − ˜ e j,n ( ¯ p ( · ) , ← n,b ) | ≤ sup τ , ¯ τ ∈ [ ¯ p ( σ ) , ← n ,σ ] | ˜ e j,n ( τ ) − ˜ e j,n ( ¯ τ ) | 1 { ¯ p ( σ ) , ← n <σ } ∨ sup τ , ¯ τ ∈ [ φ, ¯ p ( φ ) , ← n ] | ˜ e j,n ( τ ) − ˜ e j,n ( ¯ τ ) | 1 { ¯ p ( φ ) , ← n >φ } . Since the wea k limit of ( ˜ e j,n ( · )) n ≥ 1 is conti nuous at σ and φ , ¯ p ( σ ) , ← n P → σ and ¯ p ( φ ) , ← n P → φ , it follo ws that sup τ ∈ [ σ,φ ] | ˜ e j,n ( ¯ p ( · ) , ← n ) − ˜ e j,n ( ¯ p ( · ) , ← n,b ) | P → 0 and that ˜ e j,n ( ¯ p ( · ) , ← n ) − ˜ e j,n ( ¯ p ( · ) , ← n,b ) ⇒ 0 in D 1 σ ,φ . L et ˜ e n ( · ) := p k n ( ¯ p ( · ) n − ( · )) = ¯ e n ( · ) + p k n ( r n ¯ F ( u r n ( · )) − ( · )) . 38 C. Y. ROBER T By Condition (C3) , su p τ ∈ [ σ,φ ] √ k n | r n ¯ F ( u r n ( τ )) − τ | → 0. It follo w s by T h e- orem 4.1 that ˜ e n ( · ) ⇒ ¯ e ( · ) in D 1 σ ,φ . No w by using V erv aat’s lemma [ 42 ], w e get p k n ( ¯ p ( · ) , ← n − ( · )) ⇒ − ¯ e ( · ) in D 1 σ ,φ . W e deduce from the d iﬀeren tiabilit y of p ( · ) ( j ) and the ﬁ n ite increment s for- m ula that p k n ( p ( ¯ p ( · ) , ← n ) ( j ) − p ( · ) ( j )) ⇒ − h j ( · ) ¯ e ( · ) in D 1 σ ,φ . Finally , we get ˆ e j,n ( · ) = ( ˜ e j,n ( ¯ p ( · ) , ← n ) − ˜ e j,n ( ¯ p ( · ) , ← n,b )) + ˜ e j,n ( ¯ p ( · ) , ← n,b ) + p k n ( p ( ¯ p ( · ) , ← n ) ( j ) − p ( · ) ( j )) ⇒ e j ( · ) − h j ( · ) ¯ e ( · ) = ˆ e j ( · ) in D 1 σ ,φ and ( ˆ e 0 ,n ( · ) , . . . , ˆ e m,n ( · )) ⇒ (ˆ e 0 ( · ) , . . . , ˆ e m ( · )) in D m +1 σ ,φ .  Pr oof of Corollar y 4.1 . W e ﬁrst recall that a map T b et w een top o- logica l vec tor sp aces B i , i = 1 , 2, is called Hadamard diﬀerentia ble tangen- tially to some sub set S ⊂ B 1 at x ∈ B 1 if there exists a contin u ous linear map T ′ ( x ) from B 1 to B 2 suc h that T ( x + t n y n ) − T ( x ) t n → T ′ ( x ) · y for all sequ ences t n ↓ 0 and y n ∈ B 1 con v erging to y ∈ S . Note that the map Ψ j in tro duced in the pro of of Prop osition 4.1 is Hadamard diﬀerent iable tangen tially to C 2 j σ ,φ at f ∈ C 2 j σ ,φ and th at Ψ ′ j ( f ( · )) · g ( · ) =  f j +1 ( · ) (ln( f 1 ( · )) f 1 ( · )) 2 − Ψ j ( f ( · )) f 1 ( · )  g 1 ( · ) − 1 j f 1 ( · ) j − 1 X i =1 ( j − i ) f 2 j − i +1 ( · ) g i +1 ( · ) − 1 ln( f 1 ( · )) f 1 ( · ) g j +1 ( · ) − 1 j f 1 ( · ) j − 1 X i =1 if j − i +1 ( · ) g i + j +1 ( · ) . W e no w p roceed b y induction. By Theorem 4.2 , ( ˆ e 0 ,n ( · ) , . . . , ˆ e m,n ( · )) ⇒ ( ˆ e 0 ( · ) , . . . , ˆ e m ( · )) CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 39 in D m +1 σ ,φ . First, we dedu ce by th e δ -metho d (see Theorem 3.9.4 in [ 41 ]) that p k n ( χ ( · ) n (1) − π (1)) ⇒ Ψ ′ 1 ( p ( · ) (0) , p ( · ) (1)) · ( ˆ e 0 ( · ) , ˆ e 1 ( · )) = w 1 ( · ) in D 1 σ ,φ . Then ˆ d 1 ,n ( · ) = max ( − p k n π (1) , min ( p k n ( χ ( · ) n (1) − π (1)) , p k n (1 − π (1)))) ⇒ ˆ d 1 ( · ) in D 1 σ ,φ and ( ˆ e 0 ,n ( · ) , ˆ e 1 ,n ( · ) , ˆ d 1 ,n ( · )) ⇒ (ˆ e 0 ( · ) , ˆ e 1 ( · ) , ˆ d 1 ( · )) in D 3 σ ,φ . Assume that we ha ve already sho wn that ( ˆ e 0 ,n ( · ) , . . . , ˆ e j,n ( · ) , ˆ d 1 ( · ) , . . . , ˆ d j − 1 ( · )) ⇒ (ˆ e 0 ( · ) , . . . , ˆ e j ( · ) , ˆ d 1 ( · ) , . . . , ˆ d j − 1 ( · )) . The δ -method yields p k n ( χ ( · ) n ( j ) − π ( j )) ⇒ Ψ ′ j ( p ( · ) (0) , . . . , p ( · ) ( j ) , π ( · ) (1) , . . . , π ( · ) ( j − 1)) × ( ˆ e 0 ( · ) , . . . , ˆ e j ( · ) , ˆ d 1 ( · ) , . . . , ˆ d j − 1 ( · )) in D 1 σ ,φ , and a str ai ghtfo rward compu tat ion sho ws that the limit is equal to w j ( · ). Let us recall that ˆ d j,n ( · ) = max − p k n π ( j ) , min p k n ( χ ( · ) n ( j ) − π ( j )) , p k n 1 − j X i =1 π ( i ) ! − ˆ ψ j,n ( · ) !! , where ˆ ψ j,n ( · ) = P j − 1 i =1 ˆ d i,n ( · ). It follo ws that ˆ d j,n ( · ) ⇒ ˆ d j ( · ) in D 1 σ ,φ , an d ( ˆ e 0 ,n ( · ) , . . . , ˆ e j +1 ,n ( · ) , ˆ d 1 ( · ) , . . . , ˆ d j ( · )) ⇒ ( ˆ e 0 ( · ) , . . . , ˆ e j +1 ( · ) , ˆ d 1 ( · ) , . . . , ˆ d j ( · )) in D 2( j +1) σ ,φ . The induction is established and ( ˆ d 1 ,n ( · ) , . . . , ˆ d m,n ( · )) ⇒ ( ˆ d 1 ( · ) , . . . , ˆ d m ( · )) in D m σ ,φ . By the C MT , w e deduce that ( ¯ d 1 ,n , . . . , ¯ d m,n ) d →  1 φ − σ Z φ σ ˆ d 1 ( τ ) dτ , . . . , 1 φ − σ Z φ σ ˆ d m ( τ ) dτ  .  Pr oof o f Corollar y 4.2 . The assertions f ollo w fr om the δ -metho d and th e CMT.  Ac kno wledgment s. W e wo uld lik e to th ank the r eferee s and the Asso ciate Editor for their commen ts which ha ve help ed to improv e sev eral asp ects of the pap er and for dr a w in g our atten tion to the references [ 5 ] and [ 6 ]. 40 C. Y. ROBER T REFERENCES [1] Ancona-Na v arrete , M. and T a wn, J. A. (2000). A comparison of metho ds for estimating th e extremal ind ex . Extr emes 3 5–38. MR1815172 [2] Bi llingsley, P. ( 1968). Conver genc e of Pr ob abili ty Me asur es . Wiley , New Y ork. MR0233396 [3] Bi llingsley, P. (1999). Conver genc e of Pr ob ability Me asur es , 2nd ed. Wiley , New Y ork. MR1700749 [4] Bi llingsley, P. (1995). Pr ob abi l ity and Me asur e , 3rd ed. Wiley , New Y ork. MR1324786 [5] Bu chmann, B. and Gr ¨ ubel, R. (2003). Decomp ounding: An estimation problem for Poisson random sums. Ann. Statist. 31 1054–1074. MR2001642 [6] Bu chmann, B. and Gr ¨ ubel, R . (2004). D ecomp ounding P oisson rand om sums: Recursively trun cated estimates in th e d iscrete case. Ann. Inst. Statist. Math. 56 743–756. MR2126809 [7] Deo, C. M. (1973). A note on empirical p rocesses of strong-mixing sequences. Ann. Pr ob ab. 5 870–875. MR0356160 [8] Doukhan, P. (1994 ). Mixing. Pr op erties and Examples . Springer, New Y ork. MR1312160 [9] Dree s, H. (1998). O n smo oth statistical tail functionals. Sc and. J. Statist . 25 187– 210. MR1614276 [10] Drees, H. (2000). W eigh ted approxima tions of tail pro cesses for β -mixing rand om v ariables. Ann. Appl. Pr ob ab. 10 1274–1301. MR1810875 [11] Drees, H . ( 2002). T ail empirical processes un der mixing conditions. In Em piric al Pr o c ess T e chniques for Dep endent Data (H. G. Dehling, T. Mik osch and M. Sorensen, eds.) 325–342. Birkh¨ auser, Boston. MR1958788 [12] Drees, H. (2003). Extreme q uan tile estimation for dep endent data with applications to ﬁn ance. Bernoul l i 9 617–657. MR1996273 [13] Embrechts, P., Kl ¨ uppelberg, C. and Mik osch, T. (1997). Mo del ling Extr emal Events . S pringer, Berlin. MR1458613 [14] Engle, R. F. (1982). Autoregressiv e conditional heteroscedasticity with estimates of t h e v ariance of United Kingdom inﬂation. Ec onometric a 50 98 7–1008. MR0666121 [15] Ferr o, C. A. T. (2003). St atistical method s for clu sters of extreme v alues. Ph.D. thesis, Lancaster Univ. [16] Ferr o, C. A . T. and Seg e rs, J. (2003). Inference for clusters of ex treme v alues. J. R oy. Statist. So c. Ser. B 65 545–556. MR1983763 [17] Goldie, C. M. (1989). Implicit ren ew al theory and tails of solutions of random equations. Ann. Appl. Pr ob ab . 1 126–166 . MR1097468 [18] de Ha an, L. and Resnick, S. (1993). Estimating th e limiting distribution of m ulti- v ariate extremes. Comm un. Statist. Sto chatic Mo dels 9 275–309. MR1213072 [19] de Haan, L., Resnick, S., Roo tzen, H . and d e V ries, C. G. (1989). Extremal b eha viour of solutions t o a sto c hastic diﬀerence eq uation with application to ARCH p rocesses. Sto chastic Pr o c ess. Appl. 32 213–224. MR1014450 [20] Hsing, T. (1984). Poi nt pro cess es associated with extreme v alue theory . Ph.D. thesis, Dept. S tatistics , Un iv. North Carolina. [21] Hsing, T . (1991). Estimating the parameters of rare even ts. Sto chastic Pr o c ess. Appl. 37 117–139. MR1091698 [22] Hsing, T. (1993). Extremal index estimation for a weakly dep endent stationary sequence. Ann. Statist. 21 2043–2071. MR1245780 CLUSTER S IZE DI STRIBUTION OF EXTREME V ALUES 41 [23] Hsing, T. (1993). On some estimates based on sample b eha vior near high level excursions. Pr ob ab. The ory R elate d Fields 95 331–356. MR1213195 [24] Hsing, T., H ¨ usler, J. and Leadbetter, M. R. (1988). O n the ex ceedance p oin t process for a stationary sequence. Pr ob ab. The ory R elate d Fields 78 97–112. MR0940870 [25] Kesten, H. (1973). R andom diﬀerence equations and renewal theory for pro ducts of random matrices. A cta M at h. 131 207–248. MR0440724 [26] La urini , F. and T a wn, J. A. (2003). New estimators for the ext remal index and other cluster characteris tics. Extr emes 6 189–211. MR2081851 [27] Leadbetter, M. R. (1983). Extremes and lo cal dep endence in stationary sequences. Z. Wahrsch. V erw. Gebiete 65 291–306. MR0722133 [28] Leadbetter, M. R., Lindgren, G. and Rootz ´ en, H. (1983). Extr emes and R elate d Pr op erties of Ran dom Se quenc es and Pr o c esses . Springer, New Y ork. MR0691492 [29] No v ak, S. Y. (2002). Multilevel clustering of extremes. Sto chastic Pr o c ess. Appl. 97 59–75. MR1870720 [30] No v ak, S. Y. (2003). On the accuracy of multiv ariate comp ound Pois son approxi- mation. Statist. Pr ob ab. L ett. 62 35–43. MR1965369 [31] O’Brien, G. L. (1987). Extreme va lues for stationary and Marko v sequ ences. Ann. Pr ob ab. 15 281–289. MR0877604 [32] P anjer, H. H. (1981). Recursive ev aluation of a family of comp ound distributions. Astin Bul l . 12 22–26. MR0632572 [33] Perfekt, R. (1994). Extremal b ehaviour of stationary Marko v chains with applica- tions. Ann. Appl. Pr ob ab. 4 529–548. MR1272738 [34] Resnick, S. and St ˘ aric ˘ a, C . (1998). T ail estimation for dep endent data. Ann. Appl. Pr ob ab. 8 1156–1183 . MR1661160 [35] Resnick, S. an d St ˘ aric ˘ a, C. (1999). Smoothing the moment estimator of the ex- treme va lue parameter. Extr emes 1 263–293 . MR1814706 [36] R ootzen, H. (1978). Extremes of moving av erages of stable pro cesses. Ann. Pr ob ab. 6 847–869. MR0494450 [37] R ootzen, H. (2008). W eak conv ergence of the tail empirical pro cess for dep endent sequences. Sto chastic Pr o c ess. Appl. d oi: 10.1016/j.spa. 2008.03.003 . [38] Smith, R. L. (1988). A countere xample concernin g the ext remal index. A dv. in Appl . Pr ob ab. 20 681–683. MR0955511 [39] Smith, R . L. and Weissman, I. (1994). Estimating th e extremal index. J. R oy. Statist. So c. Ser. B 56 515–528. MR1278224 [40] Shao , Q.-M. and Yu, H. (1996). W eak conv ergence for w eighted empirical processes of d ependent sequ ences. Ann. Pr ob ab. 24 2098–2127. MR1415243 [41] v an der V aar t, A. W . and Wellner, J. (1996). We ak Conver genc e and Empi ric al Pr o c esses . Sprin ger, New Y ork. MR1385671 [42] Ver v aa t, W. (1972). F unctional central limit theorems for pro cess es with p ositiv e drift and their inverses . Z. Wahrsch. V erw. Gebiete 23 245–253. MR0321164 [43] Weissman, I. and No v ak, S. Y. (1998). On blo c ks and runs estimators of the ex - tremal in d ex. J. Statist. Pl ann. Infer enc e 66 281–288. MR1614480 [44] Whitt, W . (2002). Sto chastic- Pr o c ess Limi ts . Springer, New Y ork. MR 18 76437 Conser v a toire Na tional des Ar ts et Metiers Chaire de modelisa tion st a tistique Case 445 292 rue S aint Ma r tin 75003 P aris France E-mail: c hrob ert@ensae.fr

Inference for the limiting cluster size distribution of extreme values

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment