An elementary approach to extreme values theory

AN ELEMENT AR Y APPR O A CH TO EXTREME V ALUES THEOR Y Ph. Barb e CNRS, F ra nce Abstrat. This note presents a rather in tuitiv e approac h to extreme v alue theory . Th is approach was de vised mostly for p edagogic al reaso n. AMS 2000 Sub jet Classiations: Primary: 62G32. Secondary: 62E20, 60F05. Keyw ords: extreme v al ue theory , dom ain of attraction, partial attraction, geometric distribution. 1. In tro duction. The purp ose of this note is to p resent a rath e r elementary a pproach to some r esults in extreme v a lues t heory . The main p r o of was d esigned mostly for p eda gogical reasons so tha t it could be taught at a v ery in tuitiv e lev e l. In particular, the main result do es no t u s e r egular v ariation the ory or t he conc e pt of type of a distribu tion. T o recall what extreme v alue th eory is ab out, let M n b e t he maxim um of n indep en d ent real ra ndom v ariables all ha ving the same distribution function F . Extr eme v alue theo r y gre w from t he search (no w complet ed) for nece ssary and suffic ie n t cond ition for M n linearly normalize d to ha v e a non d egene rate l imiting distribu tion. In particular , o ne sa ys tha t F b elon g s to a do main of max-attrac tion if ther e exist deterministic sequen ces ( a n ) n > 1 and ( b n ) n > 1 suc h th at the distr ibution of ( M n − b n ) /a n conv erge s to a non degene rate limit. This mak es the ro ot of wh at w e c all here line ar extreme v alu e theory . In c ontrast, nonline ar extreme v alue theory seeks seque nces of deterministic a nd p ossibly nonlinear fu nctions ( g n ) n > 1 suc h that the distribut ion of g n ( M n ) conv er ges to a nonde gener ate limit as n tend s to infinity . Note th a t in th is context, it is ra ther nat ural to restrict each g n to b e mono t one, and, then with out a n y loss o f genera lit y , to b e nondecr easing, even tually b y replac ing g n b y − g n . Of essential imp o rtanc e for b ot h t he linear and nonlinear extre me v alue theo r y , the quantile func tion p ertaining to the distribution function F is d efined as F ← ( u ) = inf { x : F ( x ) > u } . 1 It is c` adl` ag, tha t is right co ntin uous with left limits, as well as nondec reasing. Note t hat in linea r extr eme v alue theory , only three p ossible limiting d ist ribution can arise. In contrast, an y non d egene r ate limiting distribut ion can arise using non linear normalizat ion. This can b e seen v ery e asily a s follo ws. Cons ide r an arbitr a ry distribut ion function G an d consider F to b e t he unifor m distribution ov er (0 , 1). Define g n ( x ) = G ← ( e − n (1 − x ) ). A dire ct calculat ion shows that the distribution of n (1 − M n ) conv er g es t o t he standa rd exp on e n tial o ne. Consequently , g n ( M n ) has limiting distribution G . 2. Nonlinear extreme v alue theory . The follo wing result c haract erizes all p ossible monoton e tran sformations g n of M n suc h that g n ( M n ) has a n o ndege n erate limiting d istribution. Theorem 2.1. L et ( g n ) n > 1 b e a se quenc e of nonde cr e asing func- tions on the r e al line. The fol lowing ar e e quivalent: (i) The distribution of g n ( M n ) c onver ges to a nonde gener ate limit. (ii) The se quenc e of functions x ∈ (0 , ∞ ) 7→ g n ◦ F ← (1 − x/n ) , n > 1 , c onver ges alm o st everywher e to a nonc onstant limit. In this c ase, writing ω for a standar d exp onential r andom variable, and h for the limiting function involve d in (ii), the limiting distribu- tion function of g n ( M n ) is that o f h ( ω ) . Mor e o ver, h is c ontinuous almost everywher e. It is e asy to see from it s pr o of that Theorem 2.1 still holds if one replaces the full sequen ce ( n ) n > 1 b y a subsequenc e ( n k ) k > 1 . Thus, the same result, consider ing now subsequen ces, a p plies to so-called partial domain o f att raction. The technique use d in the pro of also sho ws that asse rtion (ii) in T h eorem 2.1 is e quiv alent to lim ǫ → 0 g ⌊ 1 /ǫ ⌋ ◦ F ← (1 − ǫx ) (2 . 1) exists almost e verywhere and is nonc onstant on (0 , ∞ ). Because of the p ed agogical mot iv ation of this note, w e giv e a complete p r o of of The orem 2.1 a s far a s the p r obabilistic a rguments are c o ncerne d. W e will need some known auxiliary resu lts which w e state as lemmas and whose pro ofs are giv en for p edagogic al reaso ns but deferr ed to an app endix. 2 Our fir st lemma is t he so-called quantile t ransfor m which consists of the following kno wn result. Lemma 2.2. L et U b e a r andom variable having a uniform dis- tribution over (0 , 1) . The r ando m varia ble F ← ( U ) has distribution function F . The se cond lemma co llects t w o e lementary facts o n conv er gence of sequen ces of functions, t h e first asse rtion b eing not muc h more than a re stateme n t of Helly’s t h eorem (see F e ller, 1970, § VI I I.6), and the wh ole lemma b e ing exer c ise 1 3 in chapter 7 of Rudin’s ( 1986) Principles of Mathematic al Analysis . Lemma 2.3. ( i) A uniformly lo c al ly b ounde d se quenc e of nonin- cr e a sing functions has an almost everywher e c onver gent subse quenc e whose limit is c ontinuous almost everywher e. (ii) A fam ily of nonincr e asing functions which c onver ges almo st everywher e to a c o ntinuous limit c onver ges everywher e and lo c a l ly uniformly. Pro of of Theorem 2.1. Le t U n b e t he maxim um o f n indep ende n t random v ariables uniformly d istribute d on (0 , 1). A direct calcu la- tion shows th at P { U n 6 x } = x n . (2 . 2) Let ω b e a random v a riable having the sta ndard exp onential distri- bution. Note th at e − ω is unifor mly distribute d ov e r (0 , 1) . Th us, (2.2) implies t hat U n has the same distribut ion as e − ω/n . Using the quan tile t ransfor m, that is Lemma 2.2, w e see that the distribution of M n is t hat of F ← ( U n ), that is, t hat of F ← ( e − ω/n ). The refore , for g n ( M n ) t o hav e a nondeg enerat e limiting dist ribution, it is n ecessar y and sufficient th at the distribut ion of h n ( ω ) = g n ◦ F ← ( e − ω/n ) (2 . 3) conv erge s as n tend s to infin it y . The intuition b ehind our pro of is that i f t his c o n v ergenc e h o lds t hen it holds almost surely b ecause the ra ndom v ariable ω do es not de p en d o n n . Thus, w e will first consider the asse rtion the sequence ( h n ) n > 1 conv erge s almost everywhere to a limit whic h is nonco nstant on ( 0 , ∞ ) . (2 . 4) 3 Pr o of t hat (2.4) implies (i) . If ( 2 .4) holds, call h the limit of th e sequence ( h n ) n > 1 . Since lim n →∞ h n = h almost ev erywhere, the distribution of the random v ariable h n ( ω ) conv e rges t o that of h ( ω ) as n t ends to infinity . Since h is noninc reasing and is not c onstant, there exists a real num b er a such that h (0 , a ) ∩ h ( a, ∞ ) is empty . This implies that th e rando m v ariable h ( ω ) is nondeg enerat e. Pr o of t hat (i) im plies (2.4) . Let G b e t he nondeg enerat e limiting distribution in v olv e d in ( i). In ord er t o pro v e that the sequence h n defined in (2.3) conv erg es, we first show th a t it sa t isfies the assump- tions of Lemma 2.3. Note that each funct ion h n is noninc reasing. Lemma 2.4. The se quenc e ( h n ) n > 1 is lo c al ly uniform ly b ounde d on (0 , ∞ ) . Pro of. Let [ a, b ] b e a b ou n ded in terv al in (0 , ∞ ). Seeking a contradiction, assume that the sequence ( h n ) n > 1 is not b ound ed on [ a, b ]. Then, w e c an extract a subsequence ( ω k ) k > 1 in [ a, b ] and a subse quence ( n k ) k > 1 suc h th at h n k ( ω k ) tends to either + ∞ or −∞ . Assume first t h at li m k →∞ h n k ( ω k ) = + ∞ . Since h n k is nonincre asing, lim k →∞ h n k ( a ) = + ∞ . Th erefor e , for an y M p o sitiv e and any k larg e enough, 1 − e − a = P { ω 6 a } 6 P { h n k ( ω ) > h n k ( a ) } 6 P { h n k ( ω ) > M } . T aking limit as k te nds to infinity w e obtain 1 − e − a 6 1 − G ( M − ). Since M is arbitrary large, t his yields 1 − e − a 6 0, whic h is the desired contradiction. If we assume tha t lim k →∞ h n k ( ω k ) = −∞ , the n lim k →∞ h n k ( b ) = −∞ . Therefore, for an y M negative and any k large eno ugh, e − b = P { ω > b } 6 P { h n k ( ω ) 6 h n k ( b ) } 6 P { h n k ( ω ) 6 M } . T aking limit as k te n ds to infinity yield s e − b 6 G ( M ), and sinc e M is arbitr ary , e − b 6 0, which is a contradiction. F rom Lemmas 2.3 and 2.4 we d educe t h at w e can find a subse - quence h n k whic h con v erges almost ev erywhere to a limit h , and, moreov er, th is limit is no n increasing . But t hen, P { h ( ω ) 6 x } = G ( x ) . 4 It fo llo ws t hat h is unique a lmost e v erywhere and that any con- v ergent subse qu ence of ( h n ) n > 1 conv erge s to h . The n, Lemma 2 .4 implies th at the sequenc e ( h n ) n > 1 conv erge s almost ev erywhere t o h . Equivalenc e b etwe en (2.4) and The or em 2.1.ii . W e con sider the sequence of func tions ˜ h n ( ω ) = g n ◦ F ← (1 − ω /n ) . Since e − ω/n > 1 − ω /n , we see that h n > ˜ h n . F or any fixed ω a nd an y n large enou gh, e − ω/n (1 − ǫ ) 6 1 − ω /n . Therefore, for n larg e enough, ˜ h n ( ω ) > h n  ω / (1 − ǫ )  . If (ii) holds the ab ov e inequalities comparing h n and ˜ h n sho w that h  ω / (1 − ǫ )  6 lim inf n →∞ ˜ h n ( ω ) 6 lim sup n →∞ ˜ h n ( ω ) 6 h ( ω ) almost e v erywhere. If ω is a contin uit y p oint of h , the n h  ω / (1 − ǫ )  tends to h ( ω ) as ǫ tends t o 0, an d, c onsequently , ˜ h n ( ω ) conv erge s to h ( ω ) . Con v ersely , if (2.4) holds, the limiting funct ion h is mo notone and lo cally b ound ed. Hence it has a t most c o un table ma n y d iscon- tin uities and it is almost e verywhere con tinuous. The same b ou nd relating h n and ˜ h n sho w that h n conv erge s almost e verywhere to h , whic h is (ii). Equivalenc e b etwe en (2.1) a nd (ii) . Clearly , if (2 .1) holds t h en as- sertion ( ii) of Theore m 2.1 ho lds. T o prov e the conv erse implica t ion, let x b e a p o in t o f contin u ity of h such t hat lim n →∞ g n ◦ F ← (1 − x/n ) = h ( x ) . Let n b e the integer part of 1 /ǫ , so that 1 / ( n + 1) < ǫ 6 1 /n . F or an y fixed η , provided that ǫ is small eno ugh, F ← (1 − x/n ) 6 F ← (1 − ǫx ) 6 F ←  1 − ( x − η ) /n  . In partic u lar, g n ◦ F ← (1 − x/n ) 6 g ⌊ 1 /ǫ ⌋ ◦ F ← (1 − ǫx ) 6 g n ◦ F ←  1 − ( x − η ) /n  . 5 T aking limit as ǫ tends to 0 and then limit as η tends to 0 and using that x is a c ontin uity p oint of h , h ( x ) 6 lim inf ǫ → 0 g ⌊ 1 /ǫ ⌋ ◦ F ← (1 − ǫx ) 6 lim sup ǫ → 0 g ⌊ 1 /ǫ ⌋ ◦ F ← (1 − ǫx ) 6 h ( x ) . This prov e s (2.1). 3. App lication to linear extreme v alue theory . The purp ose of this se ction is t o sh ow how some classical results can b e derived from Theorem 2.1. W e mostly r e strict ourself to the follo wing result, due to de Haan (1970 ) , which c haracter izes the b elon ging t o a domains of attr action. Theorem 3.1 (de Haan, 1970) . A distribution function F b elongs to a domain of m ax-attr action if and only if for any lim ǫ → 0 F ← (1 − ǫu ) − F ← (1 − ǫ ) F ← (1 − ǫv ) − F ← (1 − ǫ ) exists (3 . 1) for almost al l u and v . Remark. Theorem 3 .1 do es not state the classica l conv er gence of t yp e result, n a mely that there a r e only three p ossible t yp es of limiting distribution . This can b e recov ere d by the follo wing known argument. F o r any re al num b er ρ , de fine the funct ion k ρ ( u ) = ( u ρ − 1 ρ if ρ 6 = 0 , log u if ρ = 0 . It can b e shown (see Bingha m, Goldie and T eugels, 1 9 89, chapter 3, or th e app endix to this pap e r which repro duc e s th eir argu ment with an extra mo notonic ity assumption whic h holds here and leads to substa ntial simplification s) th a t t he limit in ( 3.1) is necessa rily of the form k ρ ( u ) /k ρ ( v ) fo r some re a l n um b er ρ . Then, taking a n = F ← (1 − 2 /n ) − F ← (1 − 1 /n ) and b n = F ← (1 − 1 /n ) , (3 . 2) w e obtain tha t the distribu tion o f ( M n − b n ) /a n conv erge s to t h at of k ρ ( ω ) /k ρ (2). An explicit calcu lation of the limiting distribu tion 6 is then easy , and the discussion acco rding to the p osition of ρ with resp e ct to 0 (larg er, smaller or equal) yields th e c lassical t hree types. Pro of of Theorem 3.1. W e mostly present t he part of the pro of related to Theo rem 2.1. Ne c essity. Assume th at F b elongs to a domain of attraction . Consider the norming co n stants ( a n ) n > 1 and ( b n ) n > 1 , as w ell a s the function s g n ( u ) = ( u − b n ) /a n . D efine h 1 /ǫ ( x ) = g ⌊ 1 /ǫ ⌋ ◦ F ← (1 − ǫx ). Theorem 2.1.i i in its form ula t ion (2.1) a sserts t hat h 1 /ǫ conv erge s almost ev erywh e re to some function h as ǫ tend s to 0. It follo ws that for almost u, v , x, y for which h ( v ) and h ( y ) are distinct, h ( u ) − h ( x ) h ( v ) − h ( y ) = lim ǫ → 0 h 1 /ǫ ( u ) − h 1 /ǫ ( x ) h 1 /ǫ ( v ) − h 1 /ǫ ( y ) = lim ǫ → 0 F ← (1 − ǫu ) − F ← (1 − ǫx ) F ← (1 − ǫv ) − F ← (1 − ǫy ) . (3 . 3) This is n ot quite (3.1) since, a priori, we ma y not b e able to c ho o se x and y to b e 1. An extra re g ular v ariation theo retic argumen t, essentially exp laine d in B ingham, G oldie an d T eu gels (198 9, chapter 3) is then neede d. F or t he sak e of complete ness and giv en the p eda gogical nature of this note , w e dev elop this arg ument i n the app e ndix. Sufficiency. If ( 3.1) ho lds then it holds ev erywhere and lo c ally u ni- formly an d the limit is of the form k ρ ( u ) /k ρ ( v ) — see B in g ham, Goldie and T e ugels, 1989, Ch a pter 3; o r, alterna tiv ely , use the reg- ular v a r iation theo retic argument in th e app endix. T aking a n and b n as in (3.2), t h is implies th at  F ← ( e − ω/n ) − b n  /a n has a limit k ρ ( ω ) /k ρ (2) as n tends to infinity . This implies (see the r eprese nta- tion for M n in t h e pro o f of Th eorem 2.1) , that t he distribut ion of ( M n − b n ) /a n conv erge s t o a nond egene r ate limit. 4. On the maximum of geometric random v ariables. In this se c tion w e con sider t he maximum M n of n inde p end en t ra n dom v ariables all having a g eometr ic distr ibution. With the nota tion of sec tion 1 and writing ⌊·⌋ fo r the in t eger p art, the und e rlying distribution func tion is F ( t ) = (1 − p ) X 0 6 i 6 t p i = 1 − p ⌊ t +1 ⌋ 7 for some p b etw ee n 0 an d 1 . It is known ( see e .g. Resnic k, 1987, § 1.1, example following Corollary 1.6) that there a re no seque nces ( a n ) n > 1 and ( b n ) n > 1 suc h tha t th e distr ibu tion o f ( M n − b n ) /a n has a nondege nerat e limiting dist ribution. In othe r w ords, it is not p o ssible to find linear nor malizations or a seque nce of deter ministic affine function s ( g n ) n > 1 suc h th at the dist ribution of g n ( M n ) conv e rges to a no ndegen erate limit. A nat u ral qu estion is then: can we find a sequence of nonlinear funct ions ( g n ) n > 1 suc h that the distribution of g n ( M n ) ha s a nondeg e nerat e limit? T h e ne xt prop osition shows that unde r the addit ional requireme n t t hat each g n is monot one, the answ er is n egative. Henc e, in some sense, the re is no g o o d altern a tiv e to u sing subsequen c es and partial domain of att r action — see also the re mark following the pro of. The same r esult can b e o b tained in com bining theor ems 1.5.1 and 1.7.13 in Leadb ette r, Lindgren and Ro otz´ en (1983) . Prop osition 4.1. Ther e is no deterministic se quenc e of nonde- cr e a sing functions ( g n ) n > 1 such that the distribution of g n ( M n ) has a nonde gener ate lim it. Pro of. The pro of is by c on tradict ion an d re lies o n Theore m 2 .1. It also u ses the fo llo wing facts, st ated as a le mma, which is a classi- cal e xercise in an alytic num b er t heory ( see Hla wk a, Sc hoißengeier , T aschner, 1986, C h apter 2, exercise 8) an d whose p ro of is in the ap- p end ix. W e write F ( · ) for the frac tional part, that is F ( x ) = x − ⌊ x ⌋ . Lemma 4 .2. F or any p ositive r e al numb er θ , the se quenc e  F ( θ log n )  n > 1 is dense in [ 0 , 1 ] . In order to p ro v e Prop osition 3, and seeking a c ontradiction, as- sume that th e re exist s a det erministic se quence ( g n ) n > 1 of nond e - creasing func tions suc h tha t t he distribution of g n ( M n ) has a n onde- genera te limit. The orem 2.1 implies that g n ◦ F ← (1 − x/n ) h a s a limit almost everywhere, h ( x ), which is n onconst ant and n o nincrea sing . W e fir st calculat e th e quantile fu n ction F ← (1 − u ) = inf { t : p ⌊ t +1 ⌋ < u } = inf n t : ⌊ t + 1 ⌋ > log u log p o = j log u log p k . 8 In partic u lar, g n ◦ F ← (1 − x/n ) = g n j − log n log p + log x log p k . Set θ = − 1 / log p and y = log x/ log p . W e then hav e lim n →∞ g n ( ⌊ θ log n + y ⌋ ) = h ( p y ) . (4 . 1) Define the fun c tions k n ( u ) = g n ( u + ⌊ θ log n ⌋ ) and k ( y ) = h ( p y ) . Equalit y (4.1) is equiv alent to lim n →∞ k n ( ⌊ θ log n + y ⌋ − ⌊ θ log n ⌋ ) = k ( y ) . The adv an tage of t his equality compare d to (4.1) is that for fixed y the arg ument of k n remains of order 1, while t he argument of g n in (4.1) tend s to infi nit y with n . Clea rly , the argume nt of k n , t hat is, ⌊ θ log n + y ⌋ − ⌊ θ log n ⌋ , is a n integer. It is e qu al to a n in teger q if and only if ⌊ θ log n + y ⌋ = q + ⌊ θ lo g n ⌋ , that is, if q + ⌊ θ log n ⌋ 6 θ log n + y < q + ⌊ θ log n ⌋ + 1 , or, equiv alently , q ∈ F ( θ log n ) + ( y − 1 , y ] . Moreov er , if t his inequality ho lds then k n ( ⌊ θ log n + y ⌋ − ⌊ θ log n ⌋ ) = k n ( q ) , and the refore lim n →∞ k n ( q ) = h ( y ) . Since h is no nconst a n t a nd is noincrea sing, w e c an find y 1 and y 2 suc h th at y 1 < y 2 < y 1 + 1 and h ( y 2 ) < h ( y 1 ). Note t hat for any in teger n , the interv als F ( θ log n ) + ( y 1 − 1 , y 1 ] and F ( θ log n ) + ( y 2 − 1 , y 2 ] ha v e a no nempty in tersection equal to the in terv al F ( θ log n ) + ( y 2 − 1 , y 1 ]. Let ǫ b e a p ositive real n um b er such th at 2 ǫ < h ( y 1 ) − h ( y 2 ). S ince the sequen c e  F ( θ log n )  n > 1 is dense in [ 0 , 1 ], t here exists infinitely many n such th at the in tersec tions 9 F ( θ log n ) + ( y 2 − 1 , y 1 ] con tain the same in t eger q . F or th ose n sufficiently large, we t hen hav e | k n ( q ) − h ( y 1 ) | < ǫ and | k n ( q ) − h ( y 2 ) | < ǫ , whic h force s | h ( y 1 ) − h ( y 2 ) | < 2 ǫ and c ontradicts our choice of ǫ . Remark. The pro of shows in fact a litt le more, namely , t hat if t here exists a dete rministic sequenc e ( g n ) n > 1 of n ondecr easing func t ions and if there e xists a subsequence n k suc h that th e d istribution of g n k ( M n k ) con v erges to a nond e gener ate limit as k tends to infi n it y , then it is ne cessary t hat the sequence  F (log n k )  is n ot de nse in [ 0 , 1 ]. This forces the se quence ( n k ) k > 1 to av o id a se t o f the form ∪ q ∈ N ( e q [ e x , e y ]) for some 0 < x < y , and hence forces t hat seque nce to contain ga ps whic h grow a t lea st geome trically . App endix. Pro of of Lemma 2.1. If s > F ← ( U ) t hen F ( s ) > U . There fore, P { F ← ( U ) < s } 6 P { U 6 F ( s ) } = F ( s ) . Since distribut ion fun ctions are right contin uou s, t h is implies P { F ← ( U ) 6 s } 6 F ( s ) . If s < F ← ( U ) the n F ( s ) 6 U . Ther e fore, P { F ← ( U ) 6 s } = 1 − P { F ← ( U ) > s } > 1 − P { U > F ( s ) } = F ( s ) . Pro of of Lem ma 2.2. (i) A quick p r o of consists in considering that up to re p lacing nonincre asing by nondec reasing such sequence defines a se quence of measure on t he c o mpact sets [ 0 , ∞ ] as w ell as [ −∞ , 0 ] a n d use Prohor o v’s theore m ( see Billingsley , 1968 , T h eorem 6.1). A more p e destrian app roach is to sp e ll out the a r guments as follo ws. Let ( f n ) n > 1 b e a seque nce a s in th e lemma and let ( x k ) k > 1 b e a sequence of num b ers den se in the real line. Since th e sequen ce  f n ( x 1 )  n > 1 is b o u nded, w e can find an increasing funct ion ϕ 1 10 mapping N into it self such that  f ϕ 1 ( n ) ( x 1 )  n > 1 conv erge s. Sup p ose that w e construc t ed an incr easing function ϕ k from N in to itself. W e const ruct ϕ k +1 b y re quiring tha t it is incre asing, maps N in to ϕ k ( N ), that is,  ϕ k +1 ( n )  n > 1 is a subsequenc e of  ϕ k ( n )  n > 1 , and  f ϕ k +1 ( n ) ( x k +1 )  n > 1 conv erge s. Then, for an y fixed k t he sequen ce  f ϕ n ( n ) ( x k )  n > 1 conv erge s, a n d w e write f ( x k ) its limit. S ince t he function s f n are nonin c reasing, so a re the function f ϕ ( n ) ( n ) and so is f on the set ( x k ) k > 1 . Moreo v er, f is lo cally b oun ded. W e exte nd f to a fun ction f define d on the whole re al line by set ting f ( x ) = lim x k ↓ x f ( x k ) = su p { f ( x k ) : x k > x } . Since f is nonincrea sing on the set ( x k ) k > 1 , the fun c tion f is nonincre asing on the re al line. Consequently , it has le f t limit ev erywhere . It is righ t contin uo u s b ecau se if x < y < x + ǫ the n w e can find x k and x ℓ suc h that x < x k < y < x ℓ < x + ǫ , which implies f ( x k ) > f ( y ) > f ( x ℓ ); he n ce f ( x ) = lim x k ↓ x f ( x k ) > lim sup y ↓ x f ( y ) > lim inf y ↓ x f ( y ) > lim x ℓ ↓ x f ( x ℓ ) = f ( x ) . This prov e s th at f is c` adl` ag. Since it is loc ally b ounded , it has countable man y discontin uit y p oin ts. Henc e , almost every real n um b er is a contin uit y p oint of f . Let x b e a contin u it y p oint of f and let us p ro v e that  f ϕ n ( n ) ( x )  n > 1 conv erge s to f ( x ) . Inde ed, if x i < x < x k , then f ϕ n ( n ) ( x i ) > f ϕ n ( n ) ( x ) > f ϕ n ( n ) ( x k ). Thus, taking limits a s n tend s to infinit y , f ( x i ) > lim sup n →∞ f ϕ n ( n ) ( x ) > lim inf n →∞ f ϕ n ( x ) ( x ) > f ( x k ) . Since x is a contin uit y p oin t of f , taking the limits as x i and x k tend to x sh o ws that lim n →∞ f ϕ n ( n ) ( x ) = f ( x ). This implies th at the subseque n ce  f ϕ n ( n )  n > 1 conv erge s almost ev erywhere. (ii) Con sider an in terv al [ a, b ]. Let η b e a p ositive re a l n um b er. The function ∆ b eing c on tinuous, it is uniformly con tin uous o n [ a, b ]. Moreov er , since ∆ ǫ is nonincre a sing, so is ∆. Th us, we can find p oints a = a 0 < a 1 < . . . < a k = b such t hat for all 0 6 i 6 k , 0 6 ∆( a i ) − ∆( a i +1 ) 6 η and lim ǫ → 0 ∆ ǫ ( a i ) = ∆( a i ) . 11 Pro vided ǫ is sma ll enoug h, | ∆ ǫ ( a i ) − ∆( a i ) | 6 η for an y 1 6 i 6 k . Consequently , if x is b etw ee n a k and a k +1 , | ∆ ǫ ( x ) − ∆( x ) | 6 | ∆ ǫ ( x ) − ∆ ǫ ( a k +1 ) | + | ∆ ǫ ( a k +1 ) − ∆( a k +1 ) | + ∆( a k +1 ) − ∆( x ) 6 ∆ ǫ ( a k ) − ∆ ǫ ( a k +1 ) + 2 η 6 | ∆ ǫ ( a k ) − ∆( a k ) | + ∆ ǫ ( a k +1 ) − ∆( a k +1 ) | + ∆( a k ) − ∆( a k +1 ) + 2 η 6 5 η . Pro of of Lemma 4.2. Conside r an interv al [ x, y ] in [ 0 , 1 ]. Let q b e an in teger . If q + x 6 θ log n 6 q + y th en F ( θ log n ) b elongs to [ x, y ]. Such n exists if th e interv al [ e ( q + x ) /θ , e ( q + y ) /θ ] c on tains an in teger. The le n gth of t h is in terv a l is e ( q + x ) /θ ( e ( y − x ) /θ − 1 ) and ten ds to infinit y with q . Hence, this in terv al contains an integer whene v er q is large enough. Regular v ariation theoretic argument for the pro of of The- orem 3.1. W e first give the extr a a rgument nee d ed t o pro v e the necessity pa r t of Theore m 3.1. Let v and y b e su c h that h ( v ) and h ( y ) a re distinct, h 1 /ǫ ( v ) and h 1 /ǫ ( y ) conv erg e to h ( v ) and h ( y ) r esp e c tiv ely a s ǫ tend s to 0. De fine the funct ion r (1 /ǫ ) = F ← (1 − ǫv ) − F ← (1 − ǫy ) . W riting ˜ h ( u ) =  h ( u ) − h ( x )  /  h ( v ) − h ( y )  , (3.3) asse rts that lim ǫ → 0 F ← (1 − ǫu ) − F ← (1 − ǫy ) r (1 /ǫ ) = ˜ h ( u ) for almost a ll u . In pa rticular, for almost all u and w , ˜ h ( uw ) = lim ǫ → 0 F ← (1 − ǫuw ) − F ← (1 − ǫwy ) r (1 /ǫw ) r (1 /ǫw ) r (1 /ǫ ) + lim ǫ → 0 F ← (1 − ǫwy ) − F ← (1 − ǫy ) r (1 /ǫ ) . (A.1) It follo ws that lim ǫ → 0 r (1 /ǫw ) / r (1 / ǫ ) exists for almost all w . Hence r is regu larly v a rying and there exist s a r eal n u mb er ρ suc h tha t 12 lim ǫ → 0 r (1 /ǫw ) / r (1 / ǫ ) = w ρ . Then (A.1) yields the func t ional equation ˜ h ( uw ) = ˜ h ( u ) w ρ + ˜ h ( w ) . If ρ v anishes, this means ˜ h ( uw ) = ˜ h ( u ) + ˜ h ( w ) . Since ˜ h is monoton e , this forces it t o b e prop ortion al to the logarith m fu nction. If ρ do es not v anish, then, p ermuting u and w , w e obtain ˜ h ( w u ) = ˜ h ( w ) u ρ + ˜ h ( u ) . Hence, equat ing the expressions obt a ined f or h ( uw ) an d h ( w u ) , we ha v e ˜ h ( u )( w ρ − 1) = ˜ h ( w ) ( u ρ − 1) . This implies t hat the funct ion ˜ h ( u ) / ( u ρ − 1) is almost e v erywhere constant. Thus, there exists a constant c such that ˜ h = ck ρ almost ev erywhere . Again, since ˜ h is monotone, t his almost ev erywhere equality ho lds in fa ct ev erywhere . In a n y case, regard less whether ρ v anishes or n o t, w e obt ain that ˜ h = ck ρ for some constan t c . This means, sett ing c 1 = h ( x ) and c 2 = c  h ( v ) − h ( y )  , h ( u ) = c 1 + c 2 k ρ ( u ) . The function h is then continous o n t he p ositive half line. Th erefor e , Lemma 2.3.ii sho ws that h 1 /ǫ conv erge s to h ev erywhere as ǫ tends to 0. In particular, lim ǫ → 0 h 1 /ǫ ( u ) − h 1 /ǫ (1) h 1 /ǫ ( v ) − h 1 /ǫ (1) = u ρ − 1 v ρ − 1 . The same argument applies for what is neede d in th e pr o of o f the sufficiency part of Theorem 3.1, namely th at if lim ǫ → 0 F ← (1 − ǫu ) − F ← (1 − ǫx ) F ← (1 − ǫv ) − F ← (1 − ǫy ) exists for almost all u, v , x, y , t hen it exists for a ll u, v, x and y . This comes from the fact that the func tions u 7→ F ← (1 − ǫu ) − F ← (1 − ǫx ) F ← (1 − ǫv ) − F ← (1 − ǫy ) 13 are monot one and that one can take v , x and y suc h that t hese function s c on v erge for almost all u as ǫ tends to 0; hence, the conv erge nce o ccure s for a ll u a nd lo cally uniformly; p ermuting the v ariables u, v, x an d y , th e con v ergenc e also o c curs lo cally uniformly with resp ect to all u, v , x an d y . Ac kno wledgemen ts. Y ears ag o, Anne-Laure F oug` eres kept asking me que st ions a b out t h e linear normalization in asympto tic extr eme v alue theory . It is a ple asure t o ac kno wledge th at her questions a re at the ro o t of this not e, an d that m y stude n ts at the Univ ersit´ e de Cergy-P on toise d uring the spring 2008 term for prompt ing me to wr ite the pro o f in this pa p er. This note also b enefi t ed from comments, remarks and suggest ions fro m Bill McCor mic k, p recise, constru ctiv e, n umerous, neede d and welcome as alw a ys. References P . Billingsly (1968) . W ea k Con v ergenc e of Probab ilit y Measur es , Willey . N.H. Bingham, C.M. G oldie, J.L. T eugels ( 1989) . Regu lar V ariation , Cam bridge. L. de Haan (1970). On Regular V ariation and its Application to the W e ak Co n v ergence of Sa mple Extremes , Mathema tical Centre T ract s 3 2, Mathe matisch Centrum Amsterda m W. F eller (1970 ) . An Intro duction to Prob abilit y Theory and its Applications , Wiley . E. Hla wk a, J. S choißengeier , R. T aschner (1986) . Geometric and Analytic Num b er Th eory , Spring er. M.R. Leadb e tter , G. Lindgr e n, H. Roo tz´ en ( 1983). Extreme and Related Prop er ties of Ran dom Sequ ences an d Pro cesses , Springe r. S.I. Resnick (1987). Extreme V alues, Regular V a riation, and Poin t Pro ce ss , Sp r inger. W. R udin (1976). Princ iples of Mathe matical Analysis , McG ra w- Hill. Ph. Barb e 90 rue de V augirard 75006 P ARIS FRANCE philipp e.bar b e@ma t h.cnrs.fr 14