Piterbargs max-discretisation theorem for stationary vector Gaussian processes observed on different grids

PITERBAR G’S MAX-DISCRETISA TION THEOREM F OR ST A TIONAR Y VECTOR GA USSIAN PROCESSES OBSER VED ON DIFFERENT GRIDS ENKELEJD HASHOR V A AND ZHONGQUAN T AN Abstract: In this pap er w e derive Piterbarg’s max-discretisation theorem for t wo diff erent grids considering centered stationary v ector Gaussian pr o cesses. So far in the literature results in this direction hav e b een derived for the joint distribution of the max imum o f Gaussia n pro ces ses ov er [0 , T ] and over a gr id R ( δ 1 ( T )) = { k δ 1 ( T ) : k = 0 , 1 , · · · } . In this pa per we extend the recen t findings by considering additionally the max im um ov er another grid R ( δ 2 ( T )). W e deriv e the join t limiting distribution of maximum of statio nary Gauss ian vector pro cesses for differ en t choices of s uch grids by letting T → ∞ . As a by-pro duct we find that the joint limiting distr ibution of the maximum ov er different grids, which we refer to a s the Piterbar g distribution, is in the case o f weakly depe ndent G aussia n processe s a max- stable distribution. Key W ords: Piter barg’s max-dis cretisation theorem; Limiting distr ibution; Piterbarg distr ibution; Pick ands constant; Ex tremes o f G aussian processe s ; Gumbel limit law; Ber man co nditio n. AMS Cl ass ification: Primary 60F05; secondar y 60G1 5 1. Introduction Let { X ( t ) , t ≥ 0 } b e a c en tered statio nary Gaussian pro cess with contin uous sa mple paths, unit v aria nce and correla tion function r ( · ) which satisfies for some α ∈ (0 , 2] r ( t ) = 1 − C | t | α + o ( | t | α ) as t → 0 and r ( t ) < 1 for t 6 = 0 , (1) where C is some p ositive cons tant . In v ar ious a pplications only realis ations of X on a discrete time grid are po ssible. F or simplicity , in this pap er we shall consider uniform grids of p oints R ( δ ) = { k δ : k = 0 , 1 , · · · } where δ := δ ( T ) > 0 depends on the parameter T > 0 . In view of the findings o f Berma n (see [5, 7]) the maximum of X taken ov er such a discrete grid has a limiting Gu mbel distribution if lim T →∞ (2 ln T ) 1 /α δ ( T ) = D , (2) with D = ∞ and the Berman condition lim T →∞ r ( T ) ln T = r (3) holds fo r r = 0. S p ecifically , for the ma ximu m M ( δ , T ) = max i :0 ≤ iδ ≤ T X ( iδ ) over R ( δ ) ∩ [0 , T ] we hav e lim T →∞ sup x ∈ R     P { a T ( M ( δ, T ) − b δ,T ) ≤ x } − e − e − x     = 0 , Date : August 5, 2021. 1 2 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN provided that bo th (2 ) and (3 ) hold, wher e a T = √ 2 ln T , b δ,T = a T − ln( a T δ √ 2 π ) a T , T > 0 . (4) F or the maximum ov e r [0 , T ] defined thus as M ( T ) = max t ∈ [0 ,T ] X ( t ) it is well-kno wn (see e.g ., [21, 1, 2, 7, 26]) that (1) and (3 ) imply lim T →∞ sup x ∈ R     P { a T ( M ( T ) − b T ) ≤ x } − e − e − x     = 0 , (5) where b T = a T + a − 1 T ln((2 π ) − 1 / 2 C 1 /α H α a − 1+2 /α T ) (6) and H α ∈ (0 , ∞ ) deno tes Pick ands consta nt, see [24, 25, 6, 21, 2, 26, 11, 3, 14, 12, 9, 17] for more details and generalisa tions of H α . The seminal contribution [27 ] derives the joint con vergence as T → ∞ of M ( T ) a nd M ( δ, T ) showing their asymptotic independence, i.e., lim T →∞ sup x,y ∈ R     P { a T ( M ( T ) − b T ) ≤ x, a T ( M ( δ, T ) − b δ,T ≤ y } − e − e − x − e − y     = 0 . Hereafter we set B ∗ α/ 2 ( t ) := √ 2 B α/ 2 ( t ) − | t | α , t ≥ 0 with B α a standar d fractiona l Brownian motion with Hurs t index α/ 2 ∈ (0 , 1); recall that δ = δ ( T ) is given by (2). Define further for an y D > 0 H D,α = lim λ →∞ λ − 1 E n e max k ∈ N : kD ∈ [0 ,λ ] B ∗ α/ 2 ( kD ) o ∈ (0 , ∞ ) and se t (the constan t C > 0 b elow rela tes to (1)) b T ( D ) = a T + a − 1 T ln((2 π ) − 1 / 2 C 1 /α H D,α a − 1+2 /α T ) . (7) F or R ( D a − 2 /α T ) , D > 0 (in this case the grid is called Pick ands grid and δ = δ ( T ) = D a − 2 /α T ), then in view of [27], Theor em 2 the stated asymptotic independence does not hold since lim T →∞ sup x,y ∈ R     P { a T ( M ( T ) − b T ) ≤ x, a T ( M ( δ, T ) − b T ( D )) ≤ y } − e − e − x − e − y + H ln H α + x, ln H D,α + y D,α     = 0 , where t he function H x,y D,α is defined for a ny x, y ∈ R a s H x,y D,α = lim λ →∞ λ − 1 H x,y D,α ( λ ) ∈ (0 , ∞ ) , (8) with H x,y D,α ( λ ) = Z s ∈ R e s P  max t ∈ [0 ,λ ] B ∗ α/ 2 ( t ) > s + x, max k ∈ N : kD ∈ [0 , λ ] B ∗ α/ 2 ( k D ) > s + y  ds. Since it follows that f or an y w ∈ R lim x →−∞ H x,w D,α = e − w H D,α , lim y →−∞ H w, y D,α = e − w H α ∈ (0 , ∞ ) , (9) PITERBAR G’S MAX-DISCRETISA TION THEOREM 3 then Q ( x, y ) = e − e − x − e − y + H D,α (ln H α + x, ln H D,α + y ) , x, y ∈ R is a biv aria te d istribution function whic h has Gum b el margina ls Q ( z , ∞ ) = Q ( ∞ , z ) = e − e − z , z ∈ R . Moreo ver Q is a biv ariate ma x -stable distribution, which we sha ll refer to as Piterba rg distribution. This multiv a riate distribution is of s ome indep endent in teres t for statistical mo delling of dep endent multiv ariate risks. In the extreme case of a dens e gr id, whic h in the terminology of [27] means th at (2) holds for D = 0, then by Theorem 3 in [27] lim T →∞ sup x,y ∈ R     P { a T ( M ( T ) − b T ) ≤ x, a T ( M ( δ, T ) − b T ) ≤ y } − e − e − min( x,y )     = 0 th us the contin uous time and the discr ete time maxima ar e a symptotically completely dep endent . In cas e of t wo different unifor m girds R ( δ 1 ) and R ( δ 2 ) a na tural question that arises is: What is the jo in t limiting b ehaviour of M ( T ) , M ( T , δ 1 ) , M ( T , δ 2 ) f or differen t t y pes of g rids? Motiv ated b y this question, our findings this con tr ibution include: a) W e sho w that M ( T , δ 1 ) and M ( T , δ 2 ) are always asymptotically indep endent if one gr id is sparse a nd the other grid is P ick ands or dense. F ur ther , we obtain the joint limiting dis tr ibution if o ne of the g rids is Pick ands, and the other g rid is Pick ands or dense. b) The Berman condition is r e la xed by assuming tha t (3 ) h olds for some r ∈ [0 , ∞ ). When r > 0 the Gaussian pro cess X is sa id to b e strong ly dep endent , see [22, 26, 23, 32, 2 9, 8] for details on the extremes of such Gaussian pro cesses. The co n tribution [34] derives Piterbar g’s ma x-discretisatio n theorem for strong ly dep endent Gaus sian pro cesses. In applications, often modelling of the ma x im um of functionals of a Gaus sian vector pro ces s is of int erest, see e.g., [38, 4, 10]. Our results in this pap er are deriv ed fo r the mo re g eneral framew ork of Gaussia n vector pro cesse s extending the recent findings o f [31] by considering simultaneously tw o differe n t grids. This pap er highlights the role of different grids in the approximation of the maximum ov er a contin uous interv al. Our results are ther efore of interest for simulation s tudies, which was the main motiv ation of [27, 19, 2 0, 36, 37, 30, 35]. c) As a by-pro duct we show that for weakly dep endent stationary Gaussia n pro ce sses the limiting distributions are max - stable. In Extreme V a lue Theory max-stable dis tr ibutions and pro cesses are character ised in different wa ys , see e.g ., [15, 13]. In order for a mult iv aria te max-sta ble distr ibution to b e a ls o useful for s tatistical mo delling, it is imp ortant to find how that distribution approximates the maxima o f certain sequences (o r triangular ar r ays). Piter barg max- s table distributions a re ther e fore imp ortant since w e show also their usefulness in the approximations of ma x ima over different g rids. Organis ation o f t he article is as follo ws. Our main r esults are pres ent ed in the next se c tion. All the pr o ofs a re relegated to Section 3 which is f ollow ed by an Appendix. 2. Main resul ts W e sha ll inv estigate in the following the a symptotics o f maxima o ver different grids o f a cen tered sta tio nary m ultiv aria te p -dimensio nal Gaussia n pr o cess { X ( t ) , t ≥ 0 } . Each co mpo nen t X k , k ≤ p o f X is a ssumed to hav e a co nstant v a riance function equa l to 1 , contin uo us sample paths and cor relation function r kk ( t ) = 4 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN C ov ( X k ( s ) , X k ( s + t )) whic h satisfies for any index k ≤ p r kk ( t ) = 1 − C | t | α + o ( | t | α ) as t → 0 and r kk ( t ) < 1 for t 6 = 0 (10) for some po sitive constants C . Hereafter we suppose that X has jointly stationary comp onents with c ross- correla tion function r kl ( t ) = C ov ( X k ( s ) , X l ( s + t )) which do es no t dep end on s for any s, t p ositive. The strong depe ndence condition for the v ec to r Gaussia n pro ces s X reads lim T →∞ r kl ( T ) ln T = r kl ∈ [0 , ∞ ) , 1 ≤ k , l ≤ p. (11) In or der to exclude the pos s ibilit y that | X k ( t ) | = | X l ( t + t 0 ) | f or some k 6 = l , t 0 > 0 max k 6 = l sup t ∈ [0 , ∞ ) | r kl ( t ) | < 1 (12) will b e further a ssumed. F or simplicity we co nsider only tw o uniform gr ids R ( δ 1 ) and R ( δ 2 ). Recall tha t δ i , i = 1 , 2 dep end on T > 0; in the case of Pic k ands g r id we set R ( δ i ) = R ( D i a − 2 /α T ) for s ome co ns tant D i > 0 , i = 1 , 2. The vector of maxima on co ntin uous time w ill b e deno ted by M ( T ) and that with resp ect to the discrete uniform g rid R ( δ i ) , i = 1 , 2 b y M ( δ i , T ). This means that the k th comp onents of these t wo random v ector s are M k ( T ) and M k ( δ i , T ), respectively which are defined by M k ( T ) = max t ∈ [0 ,T ] X k ( t ) , M k ( δ i , T ) = max t ∈ R ( δ i ) ∩ [0 ,T ] X k ( t ) , k ≤ p. F or n otationa l simplicit y w e shall set below f M ( T ) =  a T ( M 1 ( T ) − b T ) , . . . , a T ( M p ( T ) − b T )  and f M ( δ i , T ) =  a T ( M 1 ( δ i , T ) − b δ i ,T ) , . . . , a T ( M p ( δ i , T ) − b δ i ,T )  , where b δ i ,T is defined in ( 4) if the grid R ( δ i ) is spars e, b δ i ,T = b T ( D i ) is given by (7) if w e consider a Pick ands grid R ( δ i ) = R ( D i a − 2 /α T ) and for a dense grid w e set b δ i ,T = b T with b T defined in (6). In the following x , y 1 , y 2 ∈ R p are fixed vectors and Z is a p - dimensional centered Gaussia n rando m vector with co v ariances C ov ( Z k , Z l ) = r kl √ r kk r ll , 1 ≤ l ≤ k ≤ p. (13) When r kk r ll = 0 we as sume that Z k and Z l are independent, i.e., w e shall set C ov ( Z k , Z l ) = 0 . PITERBAR G’S MAX-DISCRETISA TION THEOREM 5 The op er ations with v ectors are mean t co mpo ne nt wise, for instance x ≤ y means x k ≤ y k for an y index k ≤ p , with x k and y k the k th component of x and y , respectively . Herea fter we define p T , x , y ,δ := P n f M ( T ) ≤ x , f M ( δ i , T ) ≤ y i , i = 1 , 2 o . In the first th eorem below we discuss the case when o ne of the grids is sparse. Our results s hall es ta blish that lim T →∞ sup x , y 1 , y 2 ∈ R p     p T , x , y ,δ − E ( exp  − p X k =1 f ( x k , y k 1 , y k 2 ) e − r kk + √ 2 r kk Z k  )     = 0 , (14) where t he function f is g iven b elow explicitly for eac h particular cas e . Theorem 2.1. L et { X ( t ) , t ≥ 0 } b e a c enter e d stationary Gaussian ve ctor pr o c ess as define d ab ove and let R ( δ 1 ) b e a sp arse grid. Assume that (10), (11) and (12) hold and the Gauss ian r andom ve ctor Z has a p ositive-definite c ovarianc e matrix with elements define d in (13) . i) If R ( δ 2 ) is another sp arse grid such that R ( δ 1 ) ∩ R ( δ 2 ) = ∅ or lim T →∞ δ 1 ( T ) /δ 2 ( T ) = ∞ , then (14 ) holds with f ( x k , y k 1 , y k 2 ) = e − x k + e − y k 1 + e − y k 2 . ii) L et R ( δ 2 ) b e a sp arse grid such that R ( δ 1 ) ∩ R ( δ 2 ) = R ( δ 3 ) . If R ( δ 3 ) is a non-empty grid such that lim T →∞ ln( δ 3 ( T ) δ 1 ( T ) ) = θ 1 ∈ [0 , ∞ ) , lim T →∞ ln( δ 3 ( T ) δ 2 ( T ) ) = θ 2 ∈ [0 , ∞ ) , then (14) holds with (write θ = θ 2 − θ 1 ) f ( x k , y k 1 , y k 2 ) = e − x k + e − y k 1 + e − y k 2 − e − y k 1 − θ 1 I ( y k 1 > y k 2 + θ ) − e − y k 2 − θ 2 I ( y k 1 ≤ y k 2 + θ ) , wher e I ( · ) is the indic ator function. iii) If R ( δ 2 ) = R ( D 2 a − 2 /α T ) is a Pickands grid, then (14) holds with f ( x k , y k 1 , y k 2 ) = e − x k + e − y k 1 + e − y k 2 − H ln H α + x k , ln H D 2 ,α + y k 2 D 2 ,α . iv) If R ( δ 2 ) is a dense grid, then again (14) holds with f ( x k , y k 1 , y k 2 ) = e − y k 1 + e − min( x k ,y k 2 ) . W e consider nex t the cases that one g rid is a Pic k ands g rid a nd th e second one is either a Pic k ands or a dense grid. F or p o s itive co nstants D 1 , D 2 , λ and x, z 1 , z 2 ∈ R define (recall B ∗ α/ 2 ( t ) := √ 2 B α/ 2 ( t ) − | t | α ) H z 1 ,z 2 D 1 ,D 2 ,α ( λ ) = Z s ∈ R e s P  max k ∈ N : kD i ∈ [0 ,λ ] B ∗ α/ 2 ( k D i ) > s + z i , i = 1 , 2  ds and H x,z 1 ,z 2 D 1 ,D 2 ,α ( λ ) = Z s ∈ R e s P  max t ∈ [0 ,λ ] B ∗ α/ 2 ( t ) > s + x, max k ∈ N : kD i ∈ [0 ,λ ] B ∗ α/ 2 ( k D i ) > s + z i , i = 1 , 2  ds. 6 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN Theorem 2. 2. Under the assumptions of The or em 2.1 s u pp ose further that R ( δ 1 ) = R ( D 1 a − 2 /α T ) , D 1 > 0 is a Pickands grid. i) If R ( δ 2 ) = R ( D 2 a − 2 /α T ) , D 2 ∈ (0 , ∞ ) \ { D 1 } is also a Pickands grid, then for any x, z 1 , z 2 ∈ R H z 1 ,z 2 D 1 ,D 2 ,α = lim λ →∞ H z 1 ,z 2 D 1 ,D 2 ,α ( λ ) λ ∈ (0 , ∞ ) and H x,z 1 ,z 2 D 1 ,D 2 ,α = lim λ →∞ H x,z 1 ,z 2 D 1 ,D 2 ,α ( λ ) λ ∈ (0 , ∞ ) and further (14) holds with f given by f ( x k , y k 1 , y k 2 ) = e − x k + e − y k 1 + e − y k 2 − H ln H α + x k , ln H D 1 ,α + y k 1 D 1 ,α − H ln H α + x k , ln H D 2 ,α + y k 2 D 2 ,α − H ln H D 1 ,α + y k 1 , ln H D 2 ,α + y k 2 D 1 ,D 2 ,α + H ln H α + x k , ln H D 1 ,α + y k 1 , ln H D 2 ,α + y k 2 D 1 ,D 2 ,α . ii) If R ( δ 2 ) is a dense grid, then (14) holds with f ( x k , y k 1 , y k 2 ) = e − min( x k ,y k 2 ) + e − y k 1 − H ln H α +min( x k ,y k 2 ) , ln H D 1 ,α + y k 1 D 1 ,α . iii) If b oth R ( δ 1 ) and R ( δ 2 ) ar e dense grids, t hen again (14 ) holds with f ( x k , y k 1 , y k 2 ) = e − min( x k ,y k 1 ,y k 2 ) . Remarks : a) F rom the abov e r esults it fo llows t hat the join t con vergence sta ted t herein is determined b y the choice of the grids. The dep endence parameters r lk , l , k ≤ p determine the co v aria nce of the Gaussian random vector Z and appears explicitly in the definition of the limiting distr ibution. Clearly , if each r kk equals 0, i.e., the Berman co ndition holds for each co mpo nen t of the vector pro cess, then Z do es not app ear in a n y of the limiting results ab ove. F or such cases the maxima over a sparse grid is indep endent of that taken over a Pick ands or a dense gr id. b) Condition (9) ca n b e stated in a slightly more gener al form putting therein C k instead of C . Our results can be restated then with some obvious mo difications on the constan ts involv e d. c) In [33] a particular case of Piterba r g’s max- dis cretisation theor em was inv estig ated, which in our notation corres p onds to r kk = ∞ . Considering for simplicity p = 1, so w e assume that r 11 = ∞ , then if (1) holds with α ∈ (0 , 1] and r ( t ) = o (1) , t → ∞ a conv ex f unction, and ( r ( t ) ln t ) − 1 is monotone for large t and o (1), then f or any tw o different sparse, Pick a nds or dense gr ids R ( δ 1 ) and R ( δ 2 ) w e ha ve lim T →∞ P  a ∗ T ( M ( T ) − b ∗ T ) ≤ x, a ∗ T ( M ( δ 1 , T ) − b ∗ δ 1 ,T ) ≤ y , a ∗ T ( M ( δ 2 , T ) − b ∗ δ 2 ,T ) ≤ z  = Φ(min ( x, y , z )) (15) for an y x , y , z ∈ R as T → ∞ , where a ∗ T = 1 / p r ( T ) , b ∗ δ i ,T = p (1 − r ( T )) /r ( T ) b δ i ,T and Φ denotes the distribution function of an N (0 , 1) rando m v aria ble. The proof o f the ab ove cla im follows by Theorem 2.1 in [3 3] a nd Lemma 4.5. Consequently , for this c ase different grids do not play a r ole in the limiting distribution. Note howev e r that the noramlisatio n cons tant b ∗ δ i ,T depe nds on t he t yp e of the grid. PITERBAR G’S MAX-DISCRETISA TION THEOREM 7 iv) Set for x , y 1 , y 2 ∈ R p G ( x , y 1 , y 2 ) = E ( exp  − p X k =1 f ( x k , y k 1 , y k 2 ) e − r kk + √ 2 r kk Z k  ) , where f and Z are as in Theor em 2.1 and Theor em 2.2 . It follo ws tha t lim x →−∞ H x,y 1 ,y 2 D 1 ,D 2 ,α = H y 1 ,y 2 D 1 ,D 2 ,α , lim y 1 →−∞ ,y 2 →−∞ H x,y ,z D 1 ,D 2 ,α = e − x H α , lim x →−∞ ,y 1 →−∞ H x,y 1 ,y 2 D 1 ,D 2 ,α = lim y 1 →−∞ H y 1 ,y 2 D 1 ,D 2 ,α = H y 2 D 2 ,α = e − y 2 H D 2 ,α . Hence, using further (9) we conclude that G is a non-deg enerate m ultiv ariate distribution in R 3 p , which we refer to as the Piterbarg distribution. One impor ta n t proper t y o f G is that when r kk = 0 for a ll indices k ≤ p , then it has unit G umbel marginals Λ( x ) = e − e − x , x ∈ R . Mo r eov er, G is a max-stable distribution since ( G ( x + ln n, y 1 + ln n, y 2 + ln n )) n = G ( x , y 1 , y 2 ) , x 1 , y 1 , y 2 ∈ R p , n ∈ N . In Extreme V alue Theo ry max- stable distributions ar e imp ortant for mo delling of extremes and ra r e even ts, see e.g., [2 8, 1 5] for details. 3. Proofs In this section we present several lemmas needed for the pro of of the main results. In order to establis h Piterbarg ’s max-discretisa tion theorem for multiv a riate stationar y Gaussian pro ce sses we need to clos ely follow [27], and of c o urse to stro ngly rely o n the deep ideas and the techniques presented in [26]. First, for 1 ≤ k , l ≤ p define ρ kl ( T ) = r kl / ln T . F ollowing the former reference, w e divide the in terv al [0 , T ] on to in terv als of leng th S alternating with shorter int erv als of leng th R . Let b < a < 1 b e tw o p os itive constants, wher e b will be chosen b elow (see (29)). W e shall d enote throughout in the sequel S = T a , R = T b , T > 0 . Denote the long interv als b y S l , l = 1 , · · · , n T , and the short interv a ls by R l , l = 1 , · · · , n T where n T := [ T / ( S + R )] . (16) It will b e seen fr om the pr o ofs, that a pos sible rema ining in terv al with length differe nt than S or R plays no role in our asy mptotic consider ations; we call a lso this interv al a short in terv al. Define further S = ∪ n T l =1 S l , R = ∪ n T l =1 R l and thus [0 , T ] = S ∪ R . Our pro o fs a lso rely on the ideas of [22]; we shall construct new Gaussian pro ces ses to approximate the origina l ones. F or e a ch index k ≤ p w e define a Gaus sian pr o cess η k as η k ( t ) = Y ( j ) k ( t ) , t ∈ R j ∪ S j = [( j − 1)( S + R ) , j ( S + R )) , (17 ) 8 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN where { Y ( j ) k ( t ) , t ≥ 0 } , j = 1 , · · · , n T are indep endent copies of { X k ( t ) , t ≥ 0 } . W e construct the pro ce s ses so that η k , k = 1 , · · · , p ar e indep enden t by taking Y ( j ) k to b e indepe ndent fo r an y j and k t w o po ssible indice s. The indep endence of η k and η l implies γ kl ( s, t ) := E { η k ( s ) η l ( t ) } = 0 , k 6 = l , whereas fo r any fix ed k γ kk ( s, t ) := E { η k ( s ) η k ( t ) } =    E n Y ( i ) k ( t ) , Y ( i ) k ( s ) o = r kk ( s, t ) , if t , s ∈ R i ∪ S i , for some i ≤ n T ; E n Y ( i ) k ( t ) , Y ( j ) k ( s ) o = 0 , if t ∈ R i ∪ S i , s ∈ R j ∪ S j , for some i 6 = j ≤ n T . F or k = 1 , · · · , p define ξ T k ( t ) =  1 − ρ kk ( T )  1 / 2 η k ( t ) + ρ 1 / 2 kk ( T ) Z k , 0 ≤ t ≤ T , where Z = ( Z 1 , . . . , Z p ) is a p -dimensional centered Gaussian r andom vector intro duced in Section 2, whic h is independent o f { η k ( t ) , t ≥ 0 } , k = 1 , · · · , p . Denote b y {  kl ( s, t ) , 1 ≤ k, l ≤ p } the cov aria nce functions of { ξ T k ( t ) , 0 ≤ t ≤ T , k = 1 , · · · , p } . W e ha ve  kl ( s, t ) = E  ξ T k ( s ) ξ T l ( t )  = ρ kl ( T ) , k 6 = l and  kk ( s, t ) =    r kk ( s, t ) + (1 − r kk ( s, t )) ρ kk ( T ) , t ∈ R i ∪ S i , s ∈ R j ∪ S j , i = j ; ρ kk ( T ) , t ∈ R i ∪ S i , s ∈ R j ∪ S j , i 6 = j. F or an y ε > 0 set q ε = ε (ln T ) 1 /α . (18) F or n otationa l simplicit y w e write f M ξ ( q ε , S ) =  a T ( M ξ 1 ( q ε , S ) − b T ) , . . . , a T ( M ξp ( q ε , S ) − b T )  and f M ξ ( δ i , S ) =  a T ( M ξ 1 ( δ i , S ) − b δ i ,T ) , . . . , a T ( M ξp ( δ i , S ) − b δ i ,T )  , where M ξk ( q ε , S ) = max t ∈ R ( q ε ) ∩ S ξ T k ( t ) and b δ i ,T is defined in (4) if the grid R ( δ i ) is spa rse, b δ i ,T = b T ( D i ) is given by (7) if we consider a P ick ands grid R ( δ i ) = R ( D i a − 2 /α T ) and for a dense grid b δ i ,T = b T with b T defined in (6). W e present first four lemmas. Since their pro ofs are similar to those of Lemmas 3 .1-3.4 in [31] we sha ll not give them he r e. PITERBAR G’S MAX-DISCRETISA TION THEOREM 9 Lemma 3.1. I f R ( δ 1 ) and R ( δ 2 ) ar e sp arse or Pickands grids, then for any B > 0 ther e exits some K > 0 such that for al l x k , y ki ∈ [ − B , B ] , i = 1 , 2 , k ≤ p     P n f M ( T ) ≤ x , f M ( δ i , T ) ≤ y i , i = 1 , 2 o − P n f M ( S ) ≤ x , f M ( δ i , S ) ≤ y i , i = 1 , 2 o     ≤ K (ln T ) 1 /α − 1 / 2 T b − a holds for some 0 < b < a < 1 and al l T lar ge. In the following R ( q ε ) = R ( ε/ (ln T ) 1 /α ) denotes a P ick ands grid wher e ε > 0 and q ε is defined in ( 18). Lemma 3.2. If R ( δ 1 ) and R ( δ 2 ) ar e sp arse or Pickand s grids, t hen for any B > 0 and for al l x k , y ki ∈ [ − B , B ] , i = 1 , 2 , k ≤ p     P n f M ( S ) ≤ x , f M ( δ i , S ) ≤ y i , i = 1 , 2 o − P n f M ( q ε , S ) ≤ x , f M ( δ i , S ) ≤ y i , i = 1 , 2 o     → 0 as ε ↓ 0 . Lemma 3.3. If R ( δ 1 ) and R ( δ 2 ) ar e sp arse or Pickand s grids, t hen for any B > 0 and for al l x k , y ki ∈ [ − B , B ] , i = 1 , 2 , k ≤ p lim T →∞     P n f M ( q ε , S ) ≤ x , f M ( δ i , S ) ≤ y i , i = 1 , 2 o − P n f M ξ ( q ε , S ) ≤ x , f M ξ ( δ i , S ) ≤ y i , i = 1 , 2 o     = 0 uniformly for ε > 0 . Let in the following Φ p denote the distribution function of the p -dimensio nal Gauss ian r a ndom vector Z a nd set for η k defined in (17) c M η ( δ i , S j ) =  max t ∈ R ( δ i ) ∩S j η 1 ( t ) , · · · , max t ∈ R ( δ i ) ∩S j η p ( t )  , c M η ( S j ) =  max t ∈S j η 1 ( t ) , · · · , max t ∈S j η p ( t )  . Lemma 3.4. If R ( δ 1 ) and R ( δ 2 ) ar e sp arse or Pickands grids, then for any B > 0 for al l x k , y ki ∈ [ − B , B ] , i = 1 , 2 , k ≤ p     P n f M ξ ( q ε , S ) ≤ x , f M ξ ( δ i , S ) ≤ y i , i = 1 , 2 o − Z z ∈ R p n T Y j =1 P n c M η ( S j ) ≤ u ( x , z ) , c M η ( δ i , S j ) ≤ u ( y i , z ) , i = 1 , 2 o d Φ p ( z )     → 0 as ε ↓ 0 , wher e u ( x , z ) , u ( y i , z ) , i = 1 , 2 have c omp onen t s u ( x k , z k ) = b T + x k /a T − ρ 1 / 2 kk ( T ) z k (1 − ρ kk ( T )) 1 / 2 = x k + r kk − √ 2 r kk z k a T + b T + o ( a − 1 T ) , (19) u ( y ki , z k ) = b δ i ,T + y ki /a T − ρ 1 / 2 kk ( T ) z k (1 − ρ kk ( T )) 1 / 2 = y ki + r kk − √ 2 r kk z k a T + b δ i ,T + o ( a − 1 T ) , (20) for al l x k , y ki ∈ [ − B , B ] , i = 1 , 2 , k ≤ p . 10 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN Pro of of Theorem 2.1: Since all the limits of the probabilities in Lemmas 3 .1-3.4 are p ositive for all x k , y ki ∈ [ − B , B ] , i = 1 , 2 , k ≤ p , b y letting ε ↓ 0, we have P n f M ( T ) ≤ x , f M ( δ i , T ) ≤ y i , i = 1 , 2 o ∼ Z z ∈ R p n T Y j =1 P n c M η ( S j ) ≤ u ( x , z ) , c M η ( δ i , S j ) ≤ u ( y i , z ) , i = 1 , 2 o d Φ p ( z ) as T → ∞ . Thus, if w e can prove lim T →∞     n T Y j =1 P n c M η ( S j ) ≤ u ( x , z ) , c M η ( δ i , S j ) ≤ u ( y i , z ) , i = 1 , 2 o − e x p  − p X k =1 f ( x k , y k 1 , y k 2 ) e − r kk + √ 2 r kk z k      = 0 , (21) where f ( x k , y k 1 , y k 2 ) is defined in Theore m 2.1, then a pplying the dominated conv ergence theorem we complete the pro o f o f Theor em 2.1 for the case i ) − iii ). Define next th e ev ents A k = n max t ∈ [0 ,S ] η k ( t ) > u ( x k , z k ) o , A p + k = n max t ∈ R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) o and A 2 p + k = n max t ∈ R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k ) o , k = 1 , · · · , p. i ) Using the sta tio narity of { η k ( t ) , k = 1 , · · · , p } (we write A c k for t he complimen tary ev ent of A k ) n T Y j =1 P n c M η ( S j ) ≤ u ( x , z ) , c M η ( δ i , S j ) ≤ u ( y i , z ) , i = 1 , 2 o = ( P {∩ 3 p k =1 A c k } ) n T = exp  n T ln( P {∩ 3 p k =1 A c k } )  = exp  − n T P {∪ 3 p k =1 A k } + W n T  , where n T is defined in (16). Since lim T →∞ P {∩ 3 p k =1 A k } = 1 we ge t that the re ma inder W n T satisfies W n T = o ( n T P {∪ 3 p k =1 A k } ) , T → ∞ . Next, b y Bonferroni inequality 3 p X k =1 P {A k } ≥ P {∪ 3 p k =1 A k } ≥ 3 p X k =1 P {A k } − X 1 ≤ k u ( x k , z k ) , max t ∈ [0 ,S ] η l ( t ) > u ( x l , z l )  = X 1 ≤ k u ( x k , z k )  P  max t ∈ [0 ,S ] η l ( t ) > u ( x l , z l )  ∼ X 1 ≤ k y k 2 + θ , w e th us hav e u ( y k 1 , z k ) > u ( y k 2 , z k ) f or sufficien tly large T . F ur ther, A 10 = p X k =1 P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  = p X k =1  P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k )  + P  max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  −  1 − P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) ≤ u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) ≤ u ( y k 2 , z k )   = p X k =1  P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k )  + P  max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  −  1 − P  max t ∈R ( δ 1 ) ∩ [0 ,S ] \R ( δ 2 ) ∩ [0 ,S ] η k ( t ) ≤ u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) ≤ u ( y k 2 , z k )   = p X k =1  P  max t ∈R ( δ 1 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k )  − P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k )  + P  max t ∈R ( δ 1 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )   . By Le mma 4.2 a nd (24) we have for i = 1 , 2 as T → ∞ P  max t ∈R ( δ δ i ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k )  ∼ δ i S T ( 1 δ i − 1 δ 3 ) e − y k 1 − r kk + √ 2 r kk z k ∼ S T − 1 (1 − e − θ i ) e − y k 1 − r kk + √ 2 r kk z k , T → ∞ . F urther, applying Lemma 2 in [27] (reca ll (24)) we o bta in as T → ∞ P  max t ∈R ( δ i ) ∩ [0 ,S ] η k ( t ) > u ( y ki , z k )  ∼ S T − 1 e − y ki − r kk + √ 2 r kk z k , i = 1 , 2 . By the second assertion of Le mma 4.1, the third term is o ( T a − 1 ). Next, f or y k 1 ≤ y k 2 + θ , w e ha ve u ( y k 1 , z k ) ≤ u ( y k 2 , z k ) f or sufficien t large T . Similarly , we h av e A 10 = p X k =1 P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  = p X k =1  P  max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  − P  max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  + P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )   . PITERBAR G’S MAX-DISCRETISA TION THEOREM 13 Again, in view o f t he second ass ertion of Lemma 4.1 the third term is also o ( T a − 1 ). Consequently , A 10 = p X k =1 T a − 1 [ e − y k 1 − θ 1 I ( y k 1 > y k 2 + θ ) + e − y k 2 − θ 2 I ( y k 1 ≤ y k 2 + θ )] e − r kk + √ 2 r kk z k + o ( T a − 1 ) , T → ∞ implying that as T → ∞ n T P {∪ 3 p k =1 A k } ∼ p X k =1 ( e − x k + e − y k 1 + e − y k 2 − e − y k 1 − θ 1 I ( y k 1 > y k 2 + θ ) − e − y k 2 − θ 2 I ( y k 1 ≤ y k 2 + θ )) e − r kk + √ 2 r kk z k , which co mpletes the pro of of (21). iii ) W e pr o ceed as for the p ro of of cases i ) and ii ) using the b ound (23). By L emmas 2 and 3 in [27] and (19), (20) we obta in A 1 ∼ T a − 1 p X k =1 ( e − x k + e − y k 1 + e − y k 2 ) e − r kk + √ 2 r kk z k , T → ∞ . With s imila r a rgument as for A 2 in the pro of of case i ), we conclude that A k = o ( A 1 ) , k = 2 , 3 , 4 , 5 , 6 , 7 . F urther, Lemma 2 in [27] implies A 8 = o ( A 1 ) and Lemma 4.3 yields A 10 = o ( T a − 1 ) = o ( A 1 ) , T → ∞ . Similar argumen ts as for A 11 in th e proo f of case ii ) imply A 11 = o ( A 1 ) , T → ∞ . Borrowing the arguments of [2 6], p. 176 a nd us ing Lemma 3 in [27] it follows that A 9 = p X k =1 P  max t ∈ [0 ,S ] η k ( t ) > u ( x k , z k ) , max t ∈ R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  ∼ T a − 1 p X k =1 H ln H α + x k , ln H D 2 ,α + y k 2 D 2 ,α e − r kk + √ 2 r kk z k , T → ∞ . Consequently , as T → ∞ n T P {∪ 3 p k =1 A k } ∼ p X k =1 ( e − x k + e − y k 1 + e − y k 2 − H ln H α + x k , ln H D 2 ,α + y k 2 D 2 ,α ) e − r kk + √ 2 r kk z k , which co mpletes the pro of of the claim in (21). iv ). By Lemm a 5 in [27], we hav e     P n f M ( T ) ≤ x , f M ( δ 1 , T ) ≤ y 1 , f M ( δ 2 , T ) ≤ y 2 o − P n f M ( T ) ≤ x , f M ( δ 1 , T ) ≤ y 1 , f M ( T ) ≤ y 2 o     ≤     P n f M ( δ 2 , T ) ≤ y 2 o − P n f M ( T ) ≤ y 2 o     → 0 , T → ∞ . 14 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN Now, by Theor em 2.1 of [31], w e have P n f M ( T ) ≤ x , f M ( δ 1 , T ) ≤ y 1 , f M ( T ) ≤ y 2 o = P n f M ( T ) ≤ min( x , y 2 ) , f M ( δ 1 , T ) ≤ y 1 o → E ( exp  − p X k =1 f ( x k , y k 1 , y k 2 ) e − r kk + √ 2 r kk Z k  ) , as T → ∞ with f ( x k , y k 1 , y k 2 ) = e − min( x k ,y k 2 ) + e − y k 1 establishing the pro of.  Pro of of Theorem 2.2: i ) The limiting prop erties of the tw o constants ca n be found in Lemma 4.4. W e give the pro of of the r elation of (14). As for the pro of of Theor e m 2.1 , in view of Lemmas 3.1- 3.4 and the dominated conv er gence theorem in order to establish the pr o of we need to show that (21) holds with f ( x k , y k 1 , y k 2 ) = e − x k + e − y k 1 + e − y k 2 − H ln H α + x k , ln H D 1 ,α + y k 1 D 1 ,α − H ln H α + x k , ln H D 2 ,α + y k 2 D 2 ,α − H ln H D 1 ,α + y k 1 , ln H D 2 ,α + y k 2 D 1 ,D 2 ,α + H ln H α + x k , ln H D 1 ,α + y k 1 , ln H D 2 ,α + y k 2 D 1 ,D 2 ,α . W e p ro ceed as in the pro o f of case i i ) o f Theorem 2.1 using the bo und (2 3); we h av e thus P n ∪ 3 p k =1 A k o = 3 p X k =1 P {A k } − X 1 ≤ k,l ≤ 3 p P { A k , A l } + X 1 ≤ k,l,j ≤ 3 p P {A k , A l , A j } + X 1 ≤ k, ··· ,l ≤ 3 p P {·} =: Σ 1 − Σ 2 + Σ 3 + Σ 4 . (25) By Le mma s 2 and 3 in [2 7] and (19), (20) w e obtain that Σ 1 ∼ T a − 1 p X k =1 ( e − x k + e − y k 1 + e − y k 2 ) e − r kk + √ 2 r kk z k , T → ∞ . F urther, write Σ 2 = A 2 + A 3 + A 4 + 2 A 5 + 2 A 6 + 2 A 7 + A 8 + A 9 + A 10 , (26) where A i , i = 2 , · · · , 10 a r e defined in the pro o f of ii ) of Theor e m 2.1. Hence, with similar arguments as above A i = o ( A 1 ), i = 1 , · · · , 7 and A 8 ∼ T a − 1 p X k =1 H ln H α + x k , ln H D 1 ,α + y k 1 D 1 ,α e − r kk + √ 2 r kk z k , A 9 ∼ T a − 1 p X k =1 H ln H α + x k , ln H D 2 ,α + y k 2 D 2 ,α e − r kk + √ 2 r kk z k , A 10 ∼ T a − 1 p X k =1 H ln H D 1 ,α + y k 1 , ln H D 2 ,α + y k 2 D 1 ,D 2 e − r kk + √ 2 r kk z k PITERBAR G’S MAX-DISCRETISA TION THEOREM 15 as T → ∞ , where for the estimates of A 8 and A 9 we applied Lemma 3 in [27] a nd for the estimate of A 10 we hav e used L e mma 4.4. F urther Σ 3 = X 1 ≤ k 0 B 1 = X 1 ≤ k u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  = o ( T a − 1 ) . ii) L et R ( δ 1 ) ∩ R ( δ 2 ) = R ( δ 3 ) and lim T →∞ ln( δ 3 ( T ) δ 1 ( T ) ) = θ 1 ∈ [0 , ∞ ) , lim T →∞ ln( δ 3 ( T ) δ 2 ( T ) ) = θ 2 ∈ [0 , ∞ ) hold. If y k 1 > y k 2 + θ 2 − θ 1 , then we have for k ≤ p as T → ∞ P  max t ∈R ( δ 1 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  = o ( T a − 1 ) , wher e as if y k 1 ≤ y k 2 + θ 2 − θ 1 P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  = o ( T a − 1 ) holds. Pro of of Lem ma 4.1: The follo wing fact will be e x tensively used in the pro of. F rom assumption (10) , w e can choose a n ǫ > 0 suc h that for a ll | s − t | ≤ ǫ < 2 − 1 /α 1 2 | s − t | α ≤ 1 − r kk ( s, t ) ≤ 2 | s − t | α . (27) i ) W e fir st dea l with the case lim T →∞ δ 1 ( T ) /δ 2 ( T ) = ∞ . It is easy to chec k that p X k =1 P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  PITERBAR G’S MAX-DISCRETISA TION THEOREM 17 ≤ p X k =1  P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  + P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )   . By Le mma 2 of [27] a nd the definition o f u ( y k 2 , z k ), w e have as T → ∞ P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  ∼ S δ − 1 1 ( T ) Φ( u ( y k 2 , z k )) = C S δ − 1 1 ( T ) T − 1 δ 2 ( T ) = C T a − 1 δ 2 ( T ) δ 1 ( T ) = o ( T a − 1 ) . Now, for m, n ∈ N a nd t he ǫ chosen in (27), we hav e P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  = P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  + o ( T a − 1 ) ≤ [ S/δ 1 ]+1 X n =0 P  η k ( nδ 1 ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  + o ( T a − 1 ) ≤ [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 ≤ t ≤ S | t − nδ 1 |≤ ǫ η k ( t ) > u ( y k 2 , z k ) ) + [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 ≤ mδ 2 ≤ S | nδ 1 − mδ 2 | >ǫ η k ( mδ 2 ) > u ( y k 2 , z k ) ) + o ( T a − 1 ) =: S T , 1 + S T , 2 + o ( T a − 1 ) , where [ x ] denotes the in teg er pa rt o f x . By stationarity we hav e s e tting η ∗ nk ( t ) = η k ( nδ 1 ) + η k ( t ) S T , 1 ≤ [ S/δ 1 ]+1 X n =0 P  max nδ 1 − ǫ ≤ t ≤ nδ 1 + ǫ η ∗ nk ( t ) > u ( y k 1 , z k ) + u ( y k 2 , z k )  = C S δ 1 P  max 0 ≤ t<ǫ η ∗ 0 k ( t ) > u ( y k 1 , z k ) + u ( y k 2 , z k )  . F or t he correlation function of η ∗ 0 k ( t ) = η k (0) + η k ( t ), t ∈ [0 , ǫ ] w e have 1 − E ( η ∗ 0 k ( s ))( η ∗ 0 k ( t )) p E (( η ∗ 0 k ( s )) 2 ) E (( η ∗ 0 k ( t )) 2 ) ≤ 1 − r kk ( t − s ) 2 p 1 + r kk ( t ) p 1 + r kk ( s ) ≤ 2 | t − s | α 2 − 2 ǫ α ≤ 1 − exp( −| t − s | α ) . F urther V ar ( η ∗ 0 k ( t )) = 2 + 2 r kk ( t ) = 4 − 2 | t | α (1 + o (1)) as t → 0. Hence b y Slepian’s inequality (see e.g. Theorem 7 .4.2 of [2 1]) we h av e P  max 0 ≤ t<ǫ η ∗ 0 k ( t ) > u ( y k 1 , z k ) + u ( y k 2 , z k )  18 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN = P ( max 0 ≤ t<ǫ η ∗ 0 k ( t ) p E (( η ∗ 0 k ( t )) 2 ) q E (( η ∗ 0 k ( t )) 2 ) > u ( y k 1 , z k ) + u ( y k 2 , z k ) ) ≤ P  max 0 ≤ t<ǫ W ( t ) q E (( η ∗ 0 k ( t )) 2 ) > u ( y k 1 , z k ) + u ( y k 2 , z k )  , where W is a Gaussia n zero mean stationa ry pro ce s s with cov ariance function exp( − | t | α ), th us the condition of Theorem D.3 in [26] for the case α = β hold. By that theorem S T , 1 ≤ C S δ 1 Φ  u ( y k 1 , z k ) + u ( y k 2 , z k ) 2  . The definition of u ( y ki , z k ) , i = 1 , 2 implies th us for sparse grids [ u ( y ki , z k )] 2 = 2 ln T − ln ln T + 2 ln δ − 1 i ( T ) + O (1) . (28) Consequently , from the fact that lim T →∞ δ 1 ( T ) /δ 2 ( T ) = ∞ S T , 1 ≤ C S δ 1 ( T ) 1 √ ln T T − 1 √ ln T δ 1 / 2 1 ( T ) δ 1 / 2 2 ( T ) = C T a − 1  δ 2 ( T ) δ 1 ( T )  1 / 2 = o ( T a − 1 ) , T → ∞ . Now, let ϑ kk ( t ) = sup t ≤ s ≤ S r kk ( s ). Assumption (10) implies that ϑ kk ( ǫ ) < 1 for all T and a ny ǫ ∈ (0 , 2 − 1 /α ). Consequently , we may choos e some positive constant β kk such that β kk < 1 − ϑ kk ( ǫ ) 1 + ϑ kk ( ǫ ) < 1 for all s ufficien tly la rge T . In the following we choo s e 0 < a < b < min 1 ≤ k ≤ p β kk . (29) F or the s econd term, by stationarity and Berman’s inequality (see eg. Theorem 4 .2.1 of [2 1], Theo rem C.2 of [26]), w e ha ve S T , 2 ≤ [ S/δ 1 ]+1 X n =0 X 0 ≤ mδ 2 ≤ S | nδ 1 − mδ 2 | >ǫ P { η k ( nδ 1 ) > u ( y k 1 , z k ) , η k ( mδ 2 ) > u ( y k 2 , z k ) } ≤ [ S/δ 1 ]+1 X n =0 X 0 ≤ mδ 2 ≤ S | nδ 1 − mδ 2 | >ǫ  Φ( u ( y k 1 , z k )) Φ( u ( y k 2 , z k )) + C exp  u 2 ( y k 1 , z k ) + u 2 ( y k 2 , z k ) 2(1 + r kk ( | nδ 1 − mδ 2 | ))   ≤ S δ 1 S δ 2  Φ( u ( y k 1 , z k ))Φ( u ( y k 2 , z k )) + C exp  u 2 ( y k 1 , z k ) + u 2 ( y k 2 , z k ) 2(1 + ϑ kk ( ǫ ))   =: S T , 21 + S T , 22 . Utilising a g ain (2 8) S T , 21 ≤ C S δ 1 S δ 2 ϕ ( u ( y k 1 , z k )) u ( y k 1 , z k ) ϕ ( u ( y k 2 , z k )) u ( y k 2 , z k ) ≤ C S δ 1 S δ 2 1 ln T exp  − 1 2 u 2 ( y k 1 , z k )  exp  − 1 2 u 2 ( y k 2 , z k )  ≤ C S δ 1 S δ 2 1 ln T T − 1 (ln T ) 1 / 2 δ 1 T − 1 (ln T ) 1 / 2 δ 2 PITERBAR G’S MAX-DISCRETISA TION THEOREM 19 = C T 2( a − 1) as T → ∞ . Since u ( y ki , z k ) ∼ (2 ln T ) 1 / 2 , i = 1 , 2 S T , 22 ≤ C S δ 1 S δ 2 exp  u 2 ( y k 1 , z k ) + u 2 ( y k 2 , z k ) 2(1 + ϑ kk ( ǫ ))  ≤ C T a δ 1 T a δ 2 T − 2 1+ ϑ kk ( ǫ ) ≤ C T a − 1 T a − 1 − ϑ kk ( ǫ ) 1+ ϑ kk ( ǫ ) ( δ 1 δ 2 ) − 1 . Both (2 9) and lim T →∞ (ln T ) 1 /α δ i ( T ) = ∞ imply S T , 22 = o ( T a − 1 ) as T → ∞ . Let us consider now the case that R ( δ 1 ) ∩ R ( δ 2 ) = ∅ . Without loss of generality , we s uppo s e that u ( y k 1 , z k ) < u ( y k 2 , z k ) holds for sufficien t lar g e T . By stationarity , for m, n ∈ N and ǫ > 0 w e have P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  ≤ [ S/δ 1 ]+1 X n =0 P  η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 u ( y k 2 , z k )  ≤ [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 u ( y k 1 , z k ) ) + [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 ǫ η k ( mδ 2 ) > u ( y k 2 , z k ) ) = C S δ 1 P  η k (0) > u ( y k 1 , z k ) , max 0 u ( y k 1 , z k )  + [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 ǫ η k ( mδ 2 ) > u ( y k 2 , z k ) ) =: R T , 1 + R T , 2 . Using the well-known results for biv aria te Gaussian tail pr obability (see e.g., [1 6]) setting r = r kk ( mδ 2 ) we hav e R T , 1 ≤ C S δ 1 X 0 u ( y k 1 , z k ) , η k ( mδ 2 ) > u ( y k 1 , z k ) } = C S δ 1 X 0 0 P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  ≤ [ S/δ 1 ]+1 X n =0 P  η k ( nδ 1 ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] \R ( δ 3 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  ≤ [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 u ( y k 2 , z k ) ) + [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 ǫ η k ( mδ 2 ) > u ( y k 2 , z k ) ) ≤ C S δ 1 P  η k (0) > u ( y k 1 , z k ) , max 0 u ( y k 1 , z k )  + [ S/δ 1 ]+1 X n =0 P ( η k ( nδ 1 ) > u ( y k 1 , z k ) , max 0 ǫ η k ( mδ 2 ) > u ( y k 2 , z k ) ) =: M T , 1 + M T , 2 . Using the sa me estimates fo r R T , 1 and R T , 2 , we get that b oth M T , 1 and M T , 2 are o ( T a − 1 ). The pro of when y k 1 > y k 2 + ( θ 2 − θ 1 ) is similar. This completes the pro o f of the lemma.  The next lemm a extends Lemma 2 of [27] to the non-uniform spar se grid. Let R ( δ ) = { t 1 ( T ) < t 2 ( T ) < .... } b e a no n-uniform g rid s uc h tha t δ max := max t k ( T ) ∈ [0 ,T ] ( t k ( T ) − t k − 1 ( T )) ≤ δ 0 and δ min (ln T ) 1 /α := min t k ( T ) ∈ [0 ,T ] ( t k ( T ) − t k − 1 ( T ))(ln T ) 1 /α → ∞ as T → ∞ . Lemma 4. 2. F or S = T a , a ∈ (0 , 1) we h ave for any k ≤ p P  max t ∈R ( δ ) ∩ [0 ,S ] η k ( t ) > u T  = ♯ ( R ( δ ) ∩ [0 , S ]) Φ( u T )(1 + o (1)) as u T → ∞ , wher e ♯ ( A ) denotes the num b er of the elements of the set A . PITERBAR G’S MAX-DISCRETISA TION THEOREM 21 Pro of of Lem ma 4.2: By Bonferroni ineq uality for all T larg e (set Θ T := ♯ ( R ( δ ) ∩ [0 , S ]) and u := u T ) Θ T X l =1 P { η k ( t l ( T )) > u } ≥ P  max t ∈R ( δ ) ∩ [0 ,S ] η k ( t ) > u  ≥ Θ T X l =1 P { η k ( t l ( T )) > u } − X 1 ≤ m u, η k ( t l ( T )) > u } =: P 1 + P 2 . By the stationarity of η k P 1 = Θ T Φ( u ) , whereas fo r the s econd ter m we have for all ε > 0 sufficien tly small P 2 = X 1 ≤ m u , η k ( t l ( T )) > u } + X 1 ≤ mǫ P { η k ( t m ( T )) > u , η k ( t l ( T )) > u } =: P 21 + P 22 . Similarly as in the calculations of R T , 1 setting r = r ( t l ( T ) − t m ( T )) we have P 21 = X 1 ≤ m u, η k ( t l ( T ) − t m ( T )) > u } ≤ X 1 ≤ mǫ  Φ 2 ( u ) + C exp  u 2 1 + ϑ kk ( t l ( T ) − t m ( T ))   ≤ C Θ 2 T  Φ 2 ( u ) + C exp  u 2 1 + ϑ kk ( ǫ )   22 ENKELEJD HAS HOR V A AND ZHONGQUAN T AN Noting that Θ T ≤ S /δ min = T a /δ min and by rep eating the ca lculations for S T , 2 we obtain further P 22 = Θ T Φ( u ) o (1) as T → ∞ , which c o mpletes the pro of.  Lemma 4. 3. If R ( δ 1 ) and R ( δ 2 ) ar e sp arse and Pickands gir ds, r esp e ctively, then for k ≤ p as T → ∞ P  max t ∈R ( δ 1 ) ∩ [0 ,S ] η k ( t ) > u ( y k 1 , z k ) , max t ∈R ( δ 2 ) ∩ [0 ,S ] η k ( t ) > u ( y k 2 , z k )  = o ( T a − 1 ) . Pro of of Lem ma 4.3: Since R ( δ 1 ) and R ( δ 2 ) are spa rse and Pick a nds gir ds, resp ectively , w e ha ve lim T →∞ δ 1 ( T ) /δ 2 ( T ) = ∞ . Consequently , the pro of is similar to that of the case that lim T →∞ δ 1 ( T ) /δ 2 ( T ) = ∞ of Lemma 4.1, and ther efore we o mit further details.  Let X b e a centered s ta tionary Gaussia n pro c e s s which satis fies condition (1) (as in the In tro ductio n). F o r the pro of of Theo rem 2 .2 we sha ll determine the asymptotic behaviours, as u → ∞ , of the follo wing pro ba bilities P S ( u, x ) = P  max t ∈ R ( δ 1 ) ∩ [0 ,S ] X ( t ) > u, max t ∈ R ( δ 2 ) ∩ [0 ,S ] X ( t ) > u + x u  and P S ( u, x, y ) = P  max t ∈ R ( δ 1 ) ∩ [0 ,S ] X ( t ) > u, max t ∈ R ( δ 2 ) ∩ [0 ,S ] X ( t ) > u + x u , max t ∈ [0 ,S ] X ( t ) > u + y u  , where R ( δ 1 ) = R ( c (2 ln T ) − 1 /α ) and R ( δ 2 ) = R ( d (2 ln T ) − 1 /α ) wit h c > d > 0. F or λ ∈ ( c, ∞ ) along th e lines of the pro o f of Lemma D.1 in [26] (see a lso the pro of of Lemma 1 2 .2.3 of [21]) P λu − 2 /α ( u, x ) ∼ H 0 ,x c,d,α Φ( u ) and P λu − 2 /α ( u, x, y ) ∼ H 0 ,x,y c,d,α Φ( u ) as u → ∞ , where H 0 ,x c,d,α = lim λ →∞ 1 λ Z s ∈ R e s P  max k ∈ N : kc ∈ [0 ,λ ] B ∗ α/ 2 ( k c ) > s, max k ∈ N : kd ∈ [0 ,λ ] B ∗ α/ 2 ( k d ) > s + x  ds and H 0 ,x,y c,d,α = lim λ →∞ 1 λ Z s ∈ R e s P  max k ∈ N : kc ∈ [0 ,λ ] B ∗ α/ 2 ( k c ) > s, max k : kd ∈ [0 ,λ ] B ∗ α/ 2 ( k d ) > s + x, max t ∈ [0 ,λ ] B ∗ α/ 2 ( t ) > s + y  ds. The next result ca n b e shown a lo ng the same lin es of the proo f of Theorem D.2 in [2 6]. Lemma 4. 4. F or any x, y ∈ R we have 0 < H 0 ,x c,d,α = lim λ →∞ H 0 ,x c,d,α ( λ ) λ < ∞ and 0 < H 0 ,x,y c,d,α = lim λ →∞ H 0 ,x,y c,d,α ( λ ) λ < ∞ . F u rthermor e, for any S > 0 P S ( u, x ) ∼ S H 0 ,x c,d,α u 2 /α Φ( u ) and P S ( u, x, y ) ∼ S H 0 ,x,y c,d,α u 2 /α Φ( u ) as u → ∞ . PITERBAR G’S MAX-DISCRETISA TION THEOREM 23 Lemma 4.5. L et { Z T ,ij , 1 ≤ i ≤ p, 1 ≤ j ≤ m } , T > 0 b e a r andom matrix. Supp ose that t he fol lowing c onver genc e in distribution Z T ,j := ( Z T , 1 j , . . . , Z T ,pj ) d → ( W 1 , . . . , W p ) =: W , T → ∞ is v alid for any index j ≤ m . If further Z T ,ij ≤ Z i 1 holds almost sur ely for any index i ≤ p, 2 ≤ j ≤ m , then we have the joi nt c onver genc e in distribution ( Z T , 1 , . . . , Z T ,k ) d → ( W , . . . , W ) , T → ∞ . Pro of of Lem ma 4.5: Assume for simplicity tha t m = p = 2. By the assumptions, Lemma 2 .3 in [18] implies the convergence in distributions ( Z T , 11 , Z T , 12 ) d → ( W 1 , W 1 ) , ( Z T , 21 , Z T , 22 ) d → ( W 2 , W 2 ) , T → ∞ . 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Enkelejd Hashor v a, Dep a r tment of Act uarial Science, F acul ty of Business a nd Economics (HEC Lausanne), Univ ersity of Lausanne,, UNIL-Dorigny , 101 5 Lausanne, Switzerland E-mail addr e ss : Enkelejd.Hasho rva@unil.ch Zhongquan T an, College of Ma thema tics, Physics and In forma tion E ngineering , Jiaxing Univ ersity, Jiaxing 314001, PR China, and Dep ar tmen t of Actuarial S cience, F acul ty of Business and Economics (HEC Lausanne), University of Lausanne,, UNIL-Dorigny, 10 15 Lausanne, Switzerland, Corresponding author E-mail addr e ss : zhongquan.tan@ unil.ch