math.PR 2014-08-07 0

Extremes of Order Statistics of Stationary Processes

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th la…

Authors: Krzysztof Debicki, Enkelejd Hashorva, Lanpeng Ji

Noname man uscript No. (will be inserted b y the editor) Extremes of Order Statistics of Stationary Pro cesses Krzysztof D¸ ebic ki · Enk elejd Hashorv a · Lanp eng Ji · Chengxiu Ling Receiv ed: date / Accepted: date Abstract Let { X i ( t ) , t ≥ 0 } , 1 ≤ i ≤ n b e indep endent copies of a stationary pro cess { X ( t ) , t ≥ 0 } . F or giv en positive constan ts u, T , define the set of r th conjunctions C r,T ,u := { t ∈ [0 , T ] : X r : n ( t ) > u } with X r : n ( t ) the r th largest order statistics of X i ( t ) , t ≥ 0 , 1 ≤ i ≤ n . In n umerous applications such as brain mapping and digital comm unica tio n systems, of interest is the approximation of the pro bability that the s e t of conjunctions C r,T ,u is not empt y . Imp osing the Albin’s conditions on X , in this paper we obtain an exact asymptotic expansion of this probability as u tends to infinit y . F urther more, we establish the ta il a symptotics o f the supremum of the order statistics pro cesses of skew-Gaussian pr o cesses and a Gum b el limit theorem for the minimum order sta tis tics of statio nary Gaussia n pro cesses. As a b y-pro duct we derive a version of Li and Shao’s norma l compar ison lemma fo r the minim um and the maximum of Gaussian rando m vectors. Keyw ords Conjunction · Order statistics pro cess · Albin’s conditions · Generalized Albin constant · Sk ew-Ga ussian pro cess · Li and Shao’s no rmal compariso n lemma Mathematics Sub ject Classifi cation (2010) 60G1 0 · 60G70 Pa rtial supp ort from the Swiss National Science F oundation Pro ject 200021-140633 /1 and Marie Curi e Int ernational Researc h Staff Exchange Sch eme F ello wship wi thin the 7th Eu- ropean Communit y F ramework Programme (Grant No. RARE-318984) is kindly ack nowl- edged. The first author also ac knowledg es partial support b y N arodow e Centrum Nauki Gran t No 2013/09/B/S T1/01778 (2014-2016). K. D¸ ebic ki Mathematical Institute, U nive rsity of W roc law, pl. Grunw aldzki 2/4, 50-384 W ro c la w, Poland E. Hashorv a · L. Ji · C. Li ng Departmen t of Actuarial Science, Universit y of Lausanne, UNIL-Dorigny , 1015 Lausanne, Switzerland E-mail: chen gxiu.ling@unil. c h 2 Krzysztof D¸ ebic ki et al. 1 In tro ductio n and Main Resul t Let { X ( t ) , t ≥ 0 } b e a stationary pro cess with almost s urely (a .s .) contin uo us sample paths and denote b y X 1 , . . . , X n , n ∈ N mutually independent copies of X . Of int erest in this contribution is the r th o rder statistics pro c e ss X r : n of X 1 , . . . , X n , i.e., for any t ≥ 0 X n : n ( t ) ≤ · · · ≤ X 1: n ( t ) . (1.1) Throughout the pape r , X r : n is referred to as the r th or der statistics pr o cess generated by the proc e s s X . O r der s tatistics play a cen tra l ro le in many sta- tistical applications . Natura lly , the order statistics pro ce sses are of particular int erest in statistica l applica tions which involv e the time-dynamics. If X i ( t ) is the v alue of a ce rtain ob ject (say ima ge) i measur ed at time p oint t , and u is a fixed thresho ld, then th e s et of p oints that the r th conjunction o ccurs before some time po in t T is defined by C r,T ,u := { t ∈ [0 , T ] : X r : n ( t ) > u } . In applications it is of in terest to calculate the probability that C r,T ,u is not empt y , which is g iven by p r,T ( u ) := P ( C r,T ,u 6 = φ ) = P sup t ∈ [0 ,T ] X r : n ( t ) > u ! . (1.2) Clearly , in a n engineering context wher e X i ’s mo del some ra ndo m s ignals, p r,T ( u ) relates to th e proba bilit y that at lea st r signals ov er s ho ot the thresh- old u at s ome p oint during the time in terv al [0 , T ]. Most pr ominent statistical applications, concerned with the analysis of the s urface roughness during all machinery pro cesses and functional magnetic resonanc e imaging (FMRI) da ta, relate to the calc ulations of p r,T ( u ). A metho do logy fo r the analysis of FMRI is established in the seminal c o ntribution [3 1]. There in the authors derive a p- proximations of p n,T ( u ) by calcula ting the exp ectation of Euler characteristic of C n,T ,u for a fixed high threshold u . F or certain s mo o th Ga ussian r andom fields appr oximations of p n,T ( u ) ha ve bee n discussed in [7, 1 4, 31], where a s results for some non-Ga ussian random fields are derived in [9]. E xact asymptotic expansion of p r,T ( u ) for the clas s of stationary Gaussian proces s es X was recently derived in [15]. Obviously , the Gaussian random field cannot b e used to mo del phenomena and data sets that exhibit certain non-Gaussian c haracter is tics such as sk ew ne s s. It ar ises in man y applied-oriented fields inc luding engineering, medica l science, agricultur e and environmen tal studies; see, e.g., [1, 8, 33]. In recent y ea rs, new tec hnologies such as FMRI a nd p os itron emission tomogr aphy hav e been used to co llect data concerning the living h uman bra in as well as astroph ysics. As men tioned in the literature, these images can b e efficiently modeled b y stationary random fields. Since the exa ct ca lculation o f p r,T ( u ) is no t p ossible in genera l, in this contribution we derive a pproximations of p r,T ( u ) for u large. Extremes of order statistics of stationary pro cesses 3 F or the form ulation of our main result w e need to in tro duce Albin’s con- ditions imp osed on X as s uggested in [2, 3 , 6]. In wha t follows, let D b e a non-empty subset of R . Condition A( D ) : (Gum b el MD A and conditional limit) Suppose that X (0) has a contin uous df with infinite rig ht e ndp oint, and it is in the Gumbel max- domain of attraction (MDA ), i.e., for so me positive scaling function w ( · ) w e hav e as u → ∞ P  X (0) > u + x w ( u )  = P ( X (0) > u ) e − x (1 + o (1)) , ∀ x ∈ R . (1.3) Let q = q ( u ) satisfying lim u ↑∞ q ( u ) = 0 be a s trictly pos itive non-incr easing function. Assume tha t for any y ∈ D there ex ists a rando m pr o cess { ξ y ( t ) , t ≥ 0 } , s uch that fo r any gr id of p oints 0 < t 1 < · · · < t d < ∞ we hav e the conv er gence in distribution (denoted by d → )  w ( u )( X ( q t 1 ) − u ) , . . . , w ( u )( X ( q t d ) − u )      w ( u )( X (0) − u ) > y  d →  ξ y ( t 1 ) , . . . , ξ y ( t d )  , u → ∞ . (1.4) Condition B : (Short-lasting- e xceedance) F or all pos itive constants a, T lim N →∞ lim sup u →∞ [ T / ( aq )] X k = N P ( X ( aq k ) > u | X (0) > u ) = 0 , (1.5) where q is given a s in condition A(D) and [ x ] denotes the in teger part of x . Condition C : Supp os e that there exist p ositive constants λ 0 , ρ, b, C and d > 1 such that P  X ( q t ) > u + λ w ( u ) , X (0) ≤ u    X ( q t ) > u  ≤ C t d λ − b (1.6) holds for all u la rge and all t p os itive such that 0 < t ρ < λ < λ 0 . Here w and q are given a s in condition A(D). Here w e hav e ch osen a simpler condition C than that in [2, 3]. It has b een shown in [6] that condition C a b ove is sufficient for the v alidity o f c o ndition C given in [2, 3]; s e e a lso P rop osition 2 in [4]. Note that the Albin’s conditions A(D),B,C given ab ov e are satisfied by many well-kno wn stationary pro cesses such a s χ 2 , Γ and √ F pro cesses in [2]. A co ncr ete example is the Slepia n pro ces s, see for other examples [6, 10, 29, 30]. Howev er , showing the v alidity of these conditions r equires in g eneral significant efforts. Let ξ ( i ) 0 , i ≤ n b e indepe nden t copies of the ra ndom pro ces s ξ 0 given in co n- dition A( { 0 } ). In or der to derive the exa c t asymptotics of p r,T ( u ) w e in tro duce the following consta n ts A r := lim a ↓ 0 1 a P  sup k ≥ 1 min 1 ≤ i ≤ r ξ ( i ) 0 ( ak ) ≤ 0  , r ≤ n, (1.7) 4 Krzysztof D¸ ebic ki et al. which we r e fer to as the gener alize d A lbin c onstant s . The finiteness and pos - itiveness of it will be esta blis hed in Theorem 1.1 b elow. F or nota tional sim- plicit y we set her eafter c r,n = n ! / ( r !( n − r )!). Next, we state our pr inciple result. Theorem 1.1 L et { X r : n ( t ) , t ≥ 0 } b e the r th or der st atistics pr o c ess gener ate d by the stationary pr o c ess X . If c onditions A( { 0 } ) , B and C hold for X , then for any T > 0 , as u → ∞ , P sup t ∈ [0 ,T ] X r : n ( t ) > u ! = T A r c r,n  P ( X (0) > u )  r q ( u ) (1 + o (1)) , (1.8) wher e A r define d in (1.7) is finite and p ositive. This pap er is org anized as follo ws: In Section 2 we discuss an applicatio n of Theorem 1.1 concerning the skew-Gaussian pro cesses and then derive the Gum b el limit theorem fo r the minimum order s tatistics pro cess gener ated by a stationary Gaussian proc e ss X . All the pro ofs are pr esented in Section 3. Section 4 gives an Appendix which esta blis hes a version of Li a nd Shao’s normal comparison lemma fo r the minimum and maximum order statistics of Gaussian random vectors. 2 Sk ew-Gaussi an Pro cesses and Gumbel Limi t Theorem Throughout this se ction w e assume that { X ( t ) , t ≥ 0 } is a centered statio nary Gaussian pro cess w ith a.s. con tin uous sa mple paths and cov ariance function ρ ( · ) such that, for some α ∈ (0 , 2 ] ρ ( t ) < 1 , ∀ t > 0 and ρ ( t ) = 1 − | t | α + o ( | t | α ) , t → 0 . (2.9) It is known (see, e.g., [3, 25 , 17]) that the pr o cess X sa tisfies the assumptions of Theorem 1.1 with the pro cess ξ 0 in condition A( { 0 } ) given by ξ 0 ( t ) = √ 2 Z ( t ) − t α + E , t ≥ 0 , (2.10) where E is a unit exp onential random v a riable (rv) and { Z ( t ) , t ≥ 0 } is a (independent of E ) standar d f ractio nal Bro wnian motion (fBm) with Hurst index α/ 2 ∈ (0 , 1], i.e., Z is a cen tered Gauss ian pro cess with a .s. contin uo us sample paths and cov ar iance function Co v ( Z ( s ) , Z ( t )) = 1 2  s α + t α − | s − t | α  , s, t ≥ 0 . W e note in pa ssing that the findings of Theor em 1.1 for such X co incide with those of Theorem 2.2 in [15]. Our setup here is ho wever more general than that of the aforemen tioned pa pe r , as we demonstr ate below. Let { X i ( t ) , t ≥ Extremes of order statistics of stationary pro cesses 5 0 } , i ≤ m + 1 , m ∈ N b e independent copies of X . F o r any δ ∈ (0 , 1 ] define the skew-Gaussian pro cess ζ as ζ ( t ) = δ | X ( t ) | + p 1 − δ 2 X m +1 ( t ) , | X ( t ) | =  m X i =1 X 2 i ( t )  1 2 , t ≥ 0 . (2.11) Theorem 2.2 L et { ζ r : n ( t ) , t ≥ 0 } b e the r t h or der statist ics pr o c ess gener ate d by the skew-Gaussian pr o c ess ζ . If the stationary Gaussian pr o c ess X has c ovarianc e function ρ ( · ) which satisfies (2 .9) , then for any T > 0 , as u → ∞ , P sup t ∈ [0 ,T ] ζ r : n ( t ) > u ! = T e A r,α c r,n δ r m − r 2 r − r m/ 2 ( Γ ( m/ 2)) r u 2 α + r m − 2 r e − ru 2 2 (1 + o (1)) , (2.12) wher e e A r,α is determine d by (1.7 ) with ξ ( i ) 0 , i ≤ n b eing n inde p endent c opies of ξ 0 given in (2.1 0) , and Γ ( · ) stands for the Euler Gamma fun ction. Remarks . a) The sp ecial cas e of Theo rem 2.2, for δ = 1, coincides with that obtained in Corollar y 7.3 in [25]; see also [17, 20]. b) The Pick a nds constant H α coincides with e A 1 ,α if n = 1. It is w ell-known that H 1 = 1 and H 2 = 1 / √ π . F or other v alue s of α the r ecent con tribution [1 6] (see also the excellent monogr aph [32]) sugges ts an efficient algo rithm to simu- late H α . F or n > 1 b oth c a lculation and sim ulation o f e A r,α are op en problems. In extreme v alue analysis (see, e.g., [12, 21, 17]) it is also of interest to find some norma lizing constants a T > 0 , b T ∈ R so that the linea r norma lization of the suprem um a T  sup t ∈ [0 ,T ] X r : n ( t ) − b T  conv er ges in distribution as T → ∞ , where X r : n is the r th o rder statis tics pro c e ss genera ted by the stationary Gaussian process X . The following theorem gives a Gum b el limit result for X n : n generated by a weakly dep endent s tationary Gaussian pro cess. Theorem 2.3 L et { X n : n ( t ) , t ≥ 0 } b e the minimum or der statistics pr o c ess gener ate d by t he st ationary Gaus s ian pr o c ess X with c ovarianc e function ρ ( · ) satisfying (2 .9) . If ρ ( t ) ln t = o (1) hold s as t → ∞ , then lim T →∞ sup x ∈ R     P a T  sup t ∈ [0 ,T ] X n : n ( t ) − b T  ≤ x ! − exp  − e − x      = 0 , (2.13) wher e (set b elow D := ( n/ 2) n/ 2 − 1 /α e A n,α (2 π ) − n/ 2 ) a T = √ 2 n ln T , b T = r 2 ln T n +  1 α − n 2  ln ln T + ln D √ 2 n ln T . (2.14) 6 Krzysztof D¸ ebic ki et al. Remarks . a) It follows from the pr o of of Theore m 2 .3 that a similar result still holds for the maximum or der statistics pro cess X 1: n under the same conditio n (since (4.35) holds for the maximum). b) In several applications it is o f interest to cons ide r a ra ndom time interv al T instead of T ; s e e, e .g ., [19, 28]. A s in [28] our r e s ult in (2.13) can b e extended for random int erv a ls; w e omit that result since it can b e shown with similar arguments as in the aforementioned pap er. c) The deep con tr ibution [26] shows that b esides Gumbel limit theorems, of int erest for applica tions is the growth of E  (sup t ∈ [0 ,T ] | X n : n ( t ) | ) p  for given p > 0. In v iew of Theorem 2.2 (with m = δ = 1) and applying Lemma 4.5 in [27] we obtain lim T →∞ E  sup t ∈ [0 ,T ] | X n : n ( t ) | p 2 /n ln T  p ! = 1 . (2.15) 3 Pro o fs In this section, we present pro o fs of Theor ems 1.1, 2 .2 and 2.3. W e shall rely on the metho dolo gy developed in the seminal pap er [3]. As mentioned ther e in, chec king the Albin’s conditions for sta tionary pr o cesses is usually a hard task. In Section 3 .1 w e consider X to be a s tationary pr o cess with a.s. contin uous sample paths. In Sections 3.2, 3.3 we concentrate o n the s pec ial ca se where X is a c e n tered stationary Gaussian pro ces s with a.s. contin uous sample pa ths and cov aria nce function ρ ( · ) satisfying (2.9). 3.1 Pro of of Theorem 1.1 W e beg in w ith s ome pre liminary le mma s that will be used in the pro of o f Theorem 1.1. Unless otherwise sp ecified, { X r : n ( t ) , t ≥ 0 } deno tes the r th order statistics pro cess genera ted by the stationa ry pro cess X . The next lemma plays a key ro le throughout the pro ofs. Since its pro of is straightforward, we omit it. Reca ll tha t we de fined c r,n = n ! / ( r !( n − r )! ). Lemma 1 If X (0) has a c ont inu ous distribution function, then for any t ≥ 0 P ( X r : n ( t ) > u ) = c r,n  P ( X ( t ) > u )  r (1 + o (1)) , u → ∞ . (3.16) Lemma 2 If c ondition A( D ) holds for X , then X r : n (0) has df in the Gumb el MDA with sc aling function w r ( u ) = r w ( u ) . F urther, for any grid of p oints 0 < t 1 < · · · < t d < ∞ and al l y ∈ D we have  w r ( u )( X r : n ( q t 1 ) − u ) , . . . , w r ( u )( X r : n ( q t d ) − u )      w r ( u )( X r : n (0) − u ) > r y  d →  min 1 ≤ i ≤ r rξ ( i ) y ( t 1 ) , . . . , min 1 ≤ i ≤ r rξ ( i ) y ( t d )  , u → ∞ , (3.17) wher e ξ ( i ) y , i ≤ n ar e mu tual ly indep endent c opies o f ξ y as in c ondition A ( D ). Extremes of order statistics of stationary pro cesses 7 Pro of . First, by (1.3) and (3.1 6) P  X r : n (0) > u + x rw ( u )  = P ( X r : n (0) > u ) e − x (1 + o (1)) , x ∈ R meaning that X r : n (0) has df in the Gum b el MDA with w r ( u ) = r w ( u ). F urther, it follows fro m (1.4) that the co nv erg ence in distribution  X ∗ i ( q t 1 ) , . . . , X ∗ i ( q t d )      ( X ∗ i (0) > r y ) d →  rξ ( i ) y ( t 1 ) , . . . , r ξ ( i ) y ( t d )  (3.18) holds a s u → ∞ for all i ≤ n, y ∈ D, where X ∗ i ( t ) = w r ( u )( X i ( t ) − u ). Let further Y ∗ r ( t ) = w r ( u )  X r : n ( t ) − u  and fix a grid of points 0 < t 1 < · · · < t d < ∞ . Next, we sho w that (3.17) holds when r = n . Indeed, for an y giv en constants y 1 , . . . , y d ∈ R by (3.18) P  Y ∗ n ( q t 1 ) > y 1 , . . . , Y ∗ n ( q t d ) > y d    Y ∗ n (0) > ny  = P (min 1 ≤ i ≤ n X ∗ i ( q t j ) > y j , 1 ≤ j ≤ d, min 1 ≤ i ≤ n X ∗ i (0) > ny ) P (min 1 ≤ i ≤ n X ∗ i (0) > ny ) → P  min 1 ≤ i ≤ n nξ ( i ) y ( t 1 ) > y 1 , . . . , min 1 ≤ i ≤ n nξ ( i ) y ( t d ) > y d  (3.19) as u → ∞ . Similarly , the cla im o f (3.17) holds for all r < n if we show that, for any given consta n ts y 1 , . . . , y d ∈ R P  Y ∗ r ( q t 1 ) > y 1 , . . . , Y ∗ r ( q t d ) > y d    Y ∗ r (0) > r y  = P (min 1 ≤ i ≤ r X ∗ i ( q t j ) > y j , 1 ≤ j ≤ d, min 1 ≤ i ≤ r X ∗ i (0) > r y ) P (min 1 ≤ i ≤ r X ∗ i (0) > r y ) × (1 + Υ r ( u )) , with lim u →∞ Υ r ( u ) = 0 . (3.20) Next, we only prese nt the pro of for the case that r = n − 1 and d = 1; the other cases follow by similar arg uments. By (3.16) P  Y ∗ n − 1 (0) > ( n − 1) y  = n P  min 1 ≤ i ≤ n − 1 X ∗ i (0) > ( n − 1) y  (1 + o (1)) as u → ∞ . F urther P  Y ∗ n − 1 ( q t 1 ) > y 1 , Y ∗ n − 1 (0) > ( n − 1) y  = P  Y ∗ n − 1 ( q t 1 ) > y 1 ≥ Y ∗ n ( q t 1 ) , Y ∗ n (0) > ( n − 1) y  + P  Y ∗ n ( q t 1 ) > y 1 , Y ∗ n − 1 (0) > ( n − 1) y ≥ Y ∗ n (0)  + P  Y ∗ n − 1 ( q t 1 ) > y 1 ≥ Y ∗ n ( q t 1 ) , Y ∗ n − 1 (0) > ( n − 1) y ≥ Y ∗ n (0)  + P ( Y ∗ n ( q t 1 ) > y 1 , Y ∗ n (0) > ( n − 1) y ) =: I 1 u + I 2 u + I 3 u + I 4 u . 8 Krzysztof D¸ ebic ki et al. Since as u → ∞ P ( X ∗ n ( q t 1 ) ≤ y 1 , X ∗ n (0) > ( n − 1) y ) ≤ P  X n (0) > u + y w ( u )  = o (1) , we hav e I 1 u = n P  min 1 ≤ i ≤ n − 1 X ∗ i ( q t 1 ) > y 1 , min 1 ≤ i ≤ n − 1 X ∗ i (0) > ( n − 1) y  o (1) , and similarly I 2 u = I 1 u (1 + o (1)) . Using further the fact that P ( X ∗ n ( q t 1 ) ≤ y 1 , X ∗ n (0) ≤ ( n − 1) y ) = 1 + o (1) , u → ∞ , we hav e I 3 u = X i,i ′ ≤ n P  min 1 ≤ j ≤ n,j 6 = i X ∗ j ( q t 1 ) > y 1 , X ∗ i ( q t 1 ) ≤ y 1 , min 1 ≤ j ′ ≤ n,j ′ 6 = i ′ X ∗ j ′ (0) > ( n − 1) y , X ∗ i (0) ≤ ( n − 1) y  = n P  min 1 ≤ j ≤ n − 1 X ∗ j ( q t 1 ) > y 1 , min 1 ≤ j ′ ≤ n − 1 X ∗ j ′ (0) > ( n − 1) y  × P ( X ∗ n ( q t 1 ) ≤ y 1 , X ∗ n (0) ≤ ( n − 1) y ) + c 2 ,n P  min 1 ≤ j ≤ n − 2 X ∗ j ( q t 1 ) > y 1 , min 1 ≤ j ′ ≤ n − 2 X ∗ j ′ (0) > ( n − 1) y  × P  X ∗ n − 1 ( q t 1 ) ≤ y 1 , X ∗ n − 1 (0) > ( n − 1) y  P ( X ∗ n ( q t 1 ) > y 1 , X ∗ n (0) ≤ ( n − 1) y ) = n P  min 1 ≤ j ≤ n − 1 X ∗ j ( q t 1 ) > y 1 , min 1 ≤ j ′ ≤ n − 1 X ∗ j ′ (0) > ( n − 1) y  (1 + o (1)) . Since in view of (3.19), for k = 0 , 1 , 2 , as u → ∞ , P  min 1 ≤ j ≤ n − k X ∗ j ( q t 1 ) > y 1 , min 1 ≤ j ′ ≤ n − k X ∗ j ′ (0) > ( n − 1) y  =  P ( X (0) > u )  n − k O (1) , we conclude that I 4 u = I 3 u o (1), and further that (3.20) holds fo r r = n − 1 and d = 1. This completes the pro of. ⊓ ⊔ Lemma 3 If c ondition B is sa tisfie d by X , then for any a, T p ositive lim N →∞ lim sup u →∞ [ T / ( aq )] X k = N P ( X r : n ( aq k ) > u | X r : n (0) > u ) = 0 . (3.21) Pro of . First, since for all integers k ≥ 1 and any u positive P ( X n : n ( aq k ) > u | X n : n (0) > u ) = P ( X n : n ( aq k ) > u, X n : n (0) > u ) P ( X n : n (0) > u ) =  P ( X ( aq k ) > u | X (0) > u )  n ≤ P ( X ( aq k ) > u | X (0) > u ) Extremes of order statistics of stationary pro cesses 9 holds, condition B implies the claim for r = n . If r < n , then with similar arguments as in (3.20) we hav e for large u P  X r : n ( aq k ) > u    X r : n (0) > u  =  P  X ( aq k ) > u    X (0) > u  r (1 + Υ r ( u )) ≤ K  P  X ( aq k ) > u    X (0) > u   r holds for some K > 0, hence aga in c o ndition B establishes the pr o of. ⊓ ⊔ Lemma 4 If c ondition C is satisfie d by X with the p ar ameters as ther ein, then ther e exists some p ositive c onstant C ∗ such tha t for al l u la r ge P  X r : n ( q t ) > u + λ w ( u ) , X r : n (0) ≤ u    X r : n ( q t ) > u  ≤ C ∗ t d λ − b (3.22) holds for any t p ositive satisfyi ng 0 < t ρ < λ < λ 0 . Pro of . By condition C, for sufficiently lar ge u and C ∗ = nC, r = n P  X n : n ( q t ) > u + λ w ( u ) , X n : n (0) ≤ u    X n : n ( q t ) > u  ≤ n X i =1 P  X n : n ( q t ) > u + λ w ( u ) , X i ( q t ) ≤ u  P ( X n : n ( q t ) > u ) ≤ n X i =1 P  X i ( q t ) > u + λ w ( u ) , X i (0) ≤ u, min 1 ≤ j ≤ n,j 6 = i X j ( q t ) > u  ( P ( X ( q t ) > u )) n = n X i =1 P  X i ( q t ) > u + λ w ( u ) , X i (0) ≤ u  P ( X ( q t ) > u ) = n X i =1 P  X i ( q t ) > u + λ w ( u ) , X i (0) ≤ u    X i ( q t ) > u  ≤ C ∗ t d λ − b (3.23) holds for all t positive satisfying 0 < t ρ < λ < λ 0 . If r < n , then with similar arguments as in (3.20) P  X r : n ( q t ) > u + λ w ( u ) , X r : n (0) ≤ u  = c r,n P  min 1 ≤ i ≤ r X i ( q t ) > u + λ w ( u ) , min 1 ≤ i ≤ r X i (0) ≤ u  (1 + o (1)) holds as u → ∞ . Co ns equently , it follows f rom (3.23) that there exists some po sitive constant C ∗ such that P  X r : n ( q t ) > u + λ w ( u ) , X r : n (0) ≤ u    X r : n ( q t ) > u  ≤ C ∗ t d λ − b 10 Krzysztof D¸ ebic ki et al. holds for all t > 0 and 0 < t ρ < λ < λ 0 establishing thus the pro of. ⊓ ⊔ Pro of of Theorem 1.1. The proo f is based on an application of Theor e m 1 in [3], see also Lemma A in [6]. It follows from Lemmas 2, 3 and 4 that the conditions of Theorem 1 in [3] are satisfied, hence for any T > 0 P sup t ∈ [0 ,T ] X r : n ( t ) > u ! = T A r P ( X r : n (0) > u ) q ( u ) (1 + o (1)) , u → ∞ , where the limit in the right-hand side of (1.7) exists with A r ∈ (0 , ∞ ). Hence the pro of follows from (3.16). ⊓ ⊔ 3.2 Pro of of Theorem 2.2 In the fo llowing, we focus on the specia l case wher e the pro ce s s X is a centered stationary Gaussian pro cess with a .s . contin uous sample pa ths and cov ariance function ρ ( · ) satisfying (2 .9). Befor e pro ceeding t o the pro of of Theorem 2.2, we present four lemmas. Lemma 5 If { ζ ( t ) , t ≥ 0 } is given as in (2.1 1) , then P ( ζ (0) > u ) = δ m − 1 2 1 − m/ 2 Γ ( m/ 2) u m − 2 exp  − u 2 2  (1 + o (1)) and furt her the c onver genc e in pr ob ability | X (0) |     ζ (0) > u  p → ∞ (3.24) holds as u → ∞ . Pro of . Since | X (0) | 2 / 2 ha s Gamma ( m/ 2 , 1 ) distribution, we have P ( | X (0 ) | > u ) = 2 1 − m/ 2 Γ ( m/ 2) u m − 2 exp  − u 2 2  (1 + o (1)) , u → ∞ . Hence, Theorem 2.2 in [18] implies for any δ ∈ (0 , 1] P ( ζ (0) > u ) = δ m − 1 P ( | X (0 ) | > u ) (1 + o ( 1)) , u → ∞ . Clearly , (3.24) holds for δ = 1. Next, taking a co nstant c such tha t 1 < c < 1 / √ 1 − δ 2 for δ ∈ (0 , 1), it follows from the pro of of Lemma 2.3 in [1 8] that lim u →∞ P  (1 − c p 1 − δ 2 ) u < δ | X (0) | < cδ u    ζ (0) > u  = 1 , implying thus (3.24). Hence the pro of is complete. ⊓ ⊔ Extremes of order statistics of stationary pro cesses 11 Lemma 6 L et { ζ ( t ) , t ≥ 0 } b e given a s in (2.11) . If the c ovarianc e function ρ ( · ) of the generic stationa ry Gaussian pr o c ess X satisfies (2.9) , then for any grid o f p oints 0 < t 1 < · · · < t d < ∞ t he joint c onver genc e in distribution  u ( ζ ( q t 1 ) − u ) , . . . , u ( ζ ( q t d ) − u )     ( ζ (0) > u ) d →  ξ 0 ( t 1 ) , . . . , ξ 0 ( t d )  holds as u → ∞ , wher e t he pr o c ess ξ 0 is given by (2.10) and q = q ( u ) := u − 2 /α . Pro of . By Lemma 5, for any s ∈ R lim u →∞ P  ζ (0) > u + s u  P ( ζ (0) > u ) = e − s . Consequently , we have the conv ergence in distr ibution u ( ζ (0) − u )    ( ζ (0) − u > 0) d → E , u → ∞ , with E a unit exp onential r v. In view of Theorem 5.1 in [11], it suffices to show that as u → ∞  u ( ζ ( q t 1 ) − u ) , . . . , u ( ζ ( q t d ) − u )     ( ζ (0) = u x ) d →  √ 2 Z ( t 1 ) − t α 1 + x, . . . , √ 2 Z ( t d ) − t α d + x  , u x := u + x/u holds for all d ≥ 1 and a lmost all x > 0 with Z a standard fBm with Hurst index α/ 2. Define X ∗ i = ρ ( q t j ) X i (0) , ∆ i ( t j ) = X i ( q t j ) − X ∗ i , i ≤ m + 1 , j ≤ d. F or any u > 0 and j ≤ d , we have  u [ ζ ( q t j ) − u ] − x     ( ζ (0) = u x ) = δ u   v u u t m X i =1  X 2 i (0) + 2 X ∗ i ∆ i ( t j ) − (1 − ρ 2 ( q t j )) X 2 i (0) + ∆ 2 i ( t j )  − | X (0) |   + p 1 − δ 2 u  ∆ m +1 ( t j ) − (1 − ρ ( qt j )) X m +1 (0)  !      ( ζ (0) = u x ) =: δ A u + p 1 − δ 2 B u . Let Z i , i ≤ m + 1 b e mutually indep endent copies of Z . In view of (2.9) for s, t > 0 and i ≤ m + 1 lim u →∞ u 2 Co v ( ∆ i ( s ) , ∆ i ( t )) = s α + t α − | s − t | α = 2 Co v ( Z i ( s ) , Z i ( t )) , (3.25 ) which implies the following convergence o f finite-dimensional distributions { u∆ i ( t ) , t ≥ 0 } d → { √ 2 Z i ( t ) , t ≥ 0 } , u → ∞ , i ≤ m + 1 . 12 Krzysztof D¸ ebic ki et al. By the indep endence of ∆ i ’s and X i ’s, the Z i ’s can b e chosen to be inde- pendent of ζ (0). F urther, since ( X 1 (0) , . . . , X m +1 (0)) is a c e n tered Gauss ian random vector with N (0 , 1) indep endent comp onents, we hav e the sto chastic representation (see [13]) ( X 1 (0) , . . . , X m (0) , X m +1 (0)) d = R ( O B , I p 1 − B 2 ) , (3.26) where O = ( O 1 , . . . , O m ) is a r andom vector uniformly distr ibuted o n the unit sphere of R m . Here the rv I satisfies P ( I = ± 1) = 1 / 2, the random radius R > 0 a .s . is such that R 2 has chi-square distribution with m + 1 degrees of freedom, and the rv B is suppo rted in (0 , 1) a.s. such that B 2 has b eta distribution with parameters m/ 2 , 1 / 2. Moreover, O , I , R, B , Z i , i ≤ m + 1 are m utually indepe nden t. Consequently , using the fact th at √ x 0 + x = √ x 0 + (2 √ x 0 ) − 1 x (1 + o (1)) as x → 0, together with (3.2 5) and (3.26), w e o bta in as u → ∞ A u = P m i =1 ρ ( q t j ) X i (0)[ u∆ i ( t j )] − P m i =1 u (1 − ρ 2 ( qt j )) 2 X 2 i (0) + P m i =1 u∆ 2 i ( t j ) 2 | X (0) | × (1 + o p (1))    ( ζ (0) = u x ) = P m i =1 √ 2 X i (0) Z i ( t j ) − P m i =1 X 2 i (0) u t α j + P m i =1 Z 2 i ( t j ) u | X (0) | (1 + o p (1))    ( ζ (0) = u x ) d = √ 2 m X i =1 O i Z i ( t j ) − RB u t α j + P m i =1 Z 2 i ( t j ) uRB ! × (1 + o p (1))    ( R ( δ B + p 1 − δ 2 p 1 − B 2 I ) = u x ) B u d =  √ 2 Z m +1 ( t j ) − R √ 1 − B 2 I u t α j  (1 + o p (1))    ( ζ (0) = u x ) . Since the following sto chastic repr esentation m X i =1 O i Z i ( t j ) d = Z 1 ( t j )  m X i =1 O 2 i  1 / 2 = Z 1 ( t j ) holds, w e hav e further by (3.24) δ A u + p 1 − δ 2 B u d = √ 2  δ m X i =1 O i Z i ( t j ) + p 1 − δ 2 Z m +1 ( t j )  − R ( δ B + √ 1 − δ 2 √ 1 − B 2 I ) u t α j + P m i =1 Z 2 i ( t j ) uRB  (1 + o p (1))    ( R ( δ B + p 1 − δ 2 p 1 − B 2 I ) = u x ) d → √ 2 Z ( t j ) − t α j , u → ∞ establishing the conv e r gence f or any fixed t j > 0. The join t conv er gence in distribution for 0 < t 1 < · · · < t d < ∞ can b e sho wn with similar a rguments and is therefore omitted here. ⊓ ⊔ Extremes of order statistics of stationary pro cesses 13 Lemma 7 Under the assumptions and the notation of L emma 6, for any a, T p ositive lim sup u →∞ [ T / ( aq )] X j = N P  ζ ( aq j ) > u    ζ (0) > u  → 0 , N → ∞ . Pro of . It follows from (2.9) that for any ε > 0 sma ll enoug h 1 2 t α ≤ 1 − ρ ( t ) ≤ 2 t α , ∀ t ∈ (0 , ǫ ] . Denote b y X ( q t ) − ρ ( q t ) X (0) = ( X 1 ( q t ) − ρ ( t ) X 1 (0) , . . . , X m ( q t ) − ρ ( q t ) X m (0)), and define ζ ∗ ( q t ) ≡ δ | X ( q t ) − ρ ( q t ) X (0) | + p 1 − δ 2 ( X m +1 ( q t ) − ρ ( q t ) X m +1 (0)) . (3.27) Since X ( q t ) − ρ ( q t ) X (0) is independent of X (0 ), and X ( q t ) − ρ ( q t ) X (0) d = p 1 − ρ 2 ( q t ) X (0), we hav e that ζ ∗ ( q t ) is indep endent of ζ (0), and ζ ∗ ( q t ) d = p 1 − ρ 2 ( q t ) ζ (0 ) . Moreov er, by the tria ngle inequality ζ ∗ ( q t ) ≥  δ | X ( q t ) | + p 1 − δ 2 X m +1 ( q t )  − ρ ( qt )  δ | X (0) | + p 1 − δ 2 X m +1 (0)  = ζ ( q t ) − ρ ( qt ) ζ (0) > u (1 − ρ ( q t )) provided that ζ ( q t ) > ζ (0) > u. Therefor e P  ζ ( q t ) > u    ζ (0) > u  ≤ 2 P ( ζ ( q t ) > ζ (0) > u ) P ( ζ (0) > u ) ≤ 2 P ( ζ ∗ ( q t ) > u (1 − ρ ( qt )) , ζ (0) > u ) P ( ζ (0) > u ) = 2 P ζ (0) > u s 1 − ρ ( q t ) 1 + ρ ( q t ) ! . F urthermor e, it follows fr om Cheb yshev’s inequality and Lemma 5 that for any p > m P  ζ ( q t ) > u    ζ (0) > u  ≤    2 1+ p E ( | ζ (0) | p ) t αp/ 2 , q t ∈ (0 , ǫ ] , 2 P  ζ (0) > u q λ 2  , q t ∈ ( ǫ, T ] ≤  K p t − αp/ 2 , q t ∈ (0 , ǫ ] , K p u m − 1 − p , q t ∈ ( ǫ, T ] (3.28) is satisfied for some po sitive constant K p , where λ = 1 − sup ǫ 0 , and the second inequality is due to the fact that P ( ζ (0) > u ) ≤ C u m − 1 1 √ 2 π u exp  − u 2 2  ≤ C p u m − 1 − p , u > 0 14 Krzysztof D¸ ebic ki et al. holds for some p ositive consta n ts C and C p . Hence, if p = 2(2 /α + m − 1), then for t ≥ 1 P  ζ ( q t ) > u    ζ (0) > u  ≤ K p (1 + T α ( p − m +1) / 2 ) max( t − αp/ 2 , t − α ( p − m +1) / 2 ) ≤ C p t − 2 , q t ∈ (0 , T ] . Consequently , lim sup u →∞ [ T / ( aq )] X j = N P  ζ ( aq j ) > u    ζ (0) > u  ≤ C p ∞ X j = N ( aj ) − 2 → 0 , N → ∞ establishing the pro of. ⊓ ⊔ Lemma 8 Under the assu mptions and the notation of L emm a 6 ther e exist p ositive c onstants C , p, λ 0 , u 0 and d > 1 su ch that P  ζ ( q t ) > u + λ u , ζ (0) ≤ u  ≤ C t d λ − p P ( ζ (0) > u ) for any p ositive t satisfying 0 < t α/ 2 < λ < λ 0 and al l u > u 0 . Pro of . By (2.9) there ex is ts ǫ > 0 such that ρ ( t ) ≥ 1 2 and 1 − ρ ( t ) ≤ 2 t α for all t ∈ (0 , ǫ ]. F urther, for any t po sitive sa tisfying 0 < t α/ 2 < λ < λ 0 := min(1 / 8 , ǫ α/ 2 ) and u > 1 1 ρ ( q t ) − 1 ≤ 4 t α u 2 ≤ λ 2 u 2 . Next, for X 1 /ρ ( q t ) = ( X 1 ( q t ) − ρ − 1 ( q t ) X 1 (0) , . . . , X m ( q t ) − ρ − 1 ( q t ) X m (0)) we hav e by the triang le inequality | X ( q t ) | ≤   X 1 /ρ ( q t )   + 1 ρ ( q t ) | X (0) | . F urther, letting ζ ∗∗ ( q t ) = δ   X 1 /ρ ( q t )   + p 1 − δ 2  X m +1 ( q t ) − ρ − 1 ( q t ) X m +1 (0)  , Extremes of order statistics of stationary pro cesses 15 and b y utilising s imila r argument s as for ζ ∗ given in (3.27), w e hav e that ζ ∗∗ ( q t ) is independent of ζ ( q t ) and ζ ∗∗ ( q t ) d = p 1 − ρ 2 ( q t ) /ρ ( q t ) ζ (0). Therefore, for any t po sitive satis fying 0 < t α/ 2 < λ < λ 0 and u > 1 P  ζ ( q t ) > u + λ u , ζ (0) ≤ u    ζ ( q t ) > u  ≤ P  ζ ∗∗ ( q t ) > λ u + u − ζ (0) ρ ( q t ) , ζ (0) ≤ u    ζ ( q t ) > u  ≤ P  ζ ∗∗ ( q t ) > λ u −  1 ρ ( q t ) − 1  u    ζ ( q t ) > u  ≤ P  ζ ∗∗ ( q t ) > λ 2 u  = P ζ (0) > ρ ( q t ) p 1 − ρ 2 ( q t ) λ 2 u ! ≤ P  ζ (0) > λ 8 t α/ 2  . Consequently , by Chebyshev’s inequality for any p ositive co nstant p > 2 /α P  ζ ( q t ) > u + λ u , ζ (0) ≤ u  ≤ 8 p E ( | ζ (0) | p ) t αp/ 2 λ − p P ( ζ ( q t ) > u ) holds for any t p ositive satisfying 0 < t α/ 2 < λ < λ 0 and u large. Th us t he pro of is complete. ⊓ ⊔ Pro of of Theorem 2.2 With Lemma 5–Lemma 8, w e conclude that the cla im follows by an application of Theor em 1.1. ⊓ ⊔ 3.3 Pro of of Theorem 2.3 In view of [3, 6] or [21], we need to verify tw o additional conditions (see Lem- mas 9 and 10) for the order statistics pr o cesses genera ted by the stationar y Gaussian pro cess X . Lemma 9 Under the assumptions of The or em 2.3, we have for any c onstants a, T > 0 lim ε ↓ 0 lim sup u →∞ [ ε/ P ( X n : n (0) >u )] X j =[ T / ( aq )] P  X n : n ( aq j ) > u    X n : n (0) > u  = 0 . (3.29 ) 16 Krzysztof D¸ ebic ki et al. Pro of . Recalling that X ( t ) − ρ ( t ) X (0) is indep endent of X (0), P  X n : n ( t ) > u    X n : n (0) > u  = P  X n : n ( t ) > u, X n : n (0) > u    X n : n (0) > u  = 2 n  P  X ( t ) > X (0) > u    X (0) > u  n ≤ 2 n  P  X ( t ) − ρ ( t ) X (0) > u (1 − ρ ( t )) , X (0) > u    X (0) > u  n ≤ 2 n 1 − Φ u s 1 − | ρ ( t ) | 1 + | ρ ( t ) | !! n ≤ K u − n  1 − | ρ ( t ) | 1 + | ρ ( t ) |  − n/ 2 exp  − nu 2 2 1 − | ρ ( t ) | 1 + | ρ ( t ) |  (3.30) holds for so me p o sitive constant K a nd u large (the co nstant K b elow may b e different from line to line), here Φ ( · ) denotes the standard norma l df. Now we cho ose a function g = g ( u ) suc h that lim u →∞ g ( u ) = ∞ , | ρ ( g ( u )) | = u − 2 . It follows from u − 2 ln g ( u ) = o (1) that g ( u ) ≤ exp( ǫ ′ u 2 ) for some 0 < ǫ ′ < n/ 2(1 − | ρ ( T ) | ) / (1 + | ρ ( T ) | ) and sufficiently larg e u . Now we split the sum in (3.29) at aq j = g ( u ). The first term satisfies [ g ( u ) / ( aq )] X j =[ T / ( aq )] P  X n : n ( aq j ) > u    X n : n (0) > u  ≤ K g ( u ) aq u − n  1 − | ρ ( T ) | 1 + | ρ ( T ) |  − n/ 2 exp  − nu 2 2 1 − | ρ ( T ) | 1 + | ρ ( T ) |  ≤ K u 2 /α − n exp  ǫ ′ u 2 − nu 2 2 1 − | ρ ( T ) | 1 + | ρ ( T ) |  → 0 , u → ∞ since ǫ ′ < n/ 2(1 − | ρ ( T ) | ) / (1 + | ρ ( T ) | ). F or the remaining term we hav e [ ε/ P ( X n : n (0) >u )] X j =[ g ( u ) / ( aq )] P  X n : n ( aq j ) > u    X n : n (0) > u  ≤ K ε P ( X n : n (0) > u ) u − n  1 − u − 2 1 + u − 2  − n/ 2 exp  − nu 2 2 1 − u − 2 1 + u − 2  ≤ K ε exp  − nu 2 2  1 − u − 2 1 + u − 2 − 1  ≤ K ε, u → ∞ . Therefore, the claim follows by taking ε ↓ 0 . ⊓ ⊔ In the following lemma we shall es tablish the as ymptotic independence of X n : n ov er suitable separ ate interv a ls (see condition D ′ in [3]). In the notation used b elow e A n,α is the constant app earing in (2.12), and T = T ( u ) = (2 π ) n/ 2 e A n,α u n − 2 α exp  nu 2 2  . (3.31) Extremes of order statistics of stationary pro cesses 17 Lemma 10 Under the assumptions of The or em 2.3, if futher T = T ( u ) is define d by (3.31) and a > 0 , 0 < λ < 1 ar e given c onstants, then for any 0 ≤ s 1 < · · · < s p < t 1 < · · · < t p ′ in { aq j : j ∈ Z , 0 ≤ aq j ≤ T } with t 1 − s p ≥ λT we have lim u →∞      P   p \ i =1 { X n : n ( s i ) ≤ u } , p ′ \ j =1 { X n : n ( t j ) ≤ u }   − P p \ i =1 { X n : n ( s i ) ≤ u } ! P   p ′ \ j =1 { X n : n ( t j ) ≤ u }        = 0 . (3.32) Pro of . First, taking logar ithms on b oth sides of (3.31) w e obtain ln T = nu 2 2 +  n − 2 α  ln u + ln (2 π ) n/ 2 e A n,α ! , which together with u 2 = (2 /n ) ln T (1 + o (1)) implies that u 2 = 2 ln T n +  2 nα − 1  ln ln T + ln  n 2  1 − 2 nα ( e A n,α ) 2 n 2 π (1 + o (1)) (3.33) as T → ∞ . F urther, define (hereafter I {·} denotes the indicator function) X ij = X j ( s i ) I { i ≤ p } + X j ( t i − p ) I { p < i ≤ p + p ′ } , 1 ≤ i ≤ p + p ′ , 1 ≤ j ≤ n, and { Y ij , 1 ≤ i ≤ p, 1 ≤ j ≤ n } d = { X ij , 1 ≤ i ≤ p, 1 ≤ j ≤ n } , indep endent of { Y ij , p + 1 ≤ i ≤ p + p ′ , 1 ≤ j ≤ n } d = { X ij , p + 1 ≤ i ≤ p + p ′ , 1 ≤ j ≤ n } . Applying Lemma 12 with X i ( n ) = X n : n ( s i ) I { i ≤ p } + X n : n ( t i − p ) I { p < i ≤ p + p ′ } (see the Appendix), using similar a rguments as in L e mma 8.2.4 in [2 1] we obtain that the left-hand side of (3.32) is b ounded fro m a bove by K u − 2( n − 1)  T q  X λT ≤ t j − s i ≤ T e − nu 2 1+ | ρ ( t j − s i ) | Z | ρ ( t j − s i ) | 0 (1 + | h | ) 2( n − 1) (1 − h 2 ) n/ 2 dh ≤ K u − 2( n − 1)  T q  X λT ≤ aq j ≤ T | ρ ( aq j ) | e − nu 2 1+ | ρ ( aqj ) | , for larg e u, where K is some po sitive constant. The rest of the pro of co nsists of the same arguments as that o f Lemma 12.3.1 in [21] b y using (3.33) and the Berman’s condition ρ ( t ) ln t = o (1). Hence the pro of is c o mplete. ⊓ ⊔ Pro of of Theorem 2.3 Since Theorem 2.2 and Lemmas 9 and 10 hold for the n th orde r statistics proces s X n : n , in view of Lemma B in [6] w e ha ve for T = T ( u ) defined as in ( 3.31) lim u →∞ P sup t ∈ [0 ,T ( u )] X n : n ( t ) ≤ u + x nu ! = exp  − e − x  , x ∈ R . Hence the pro of follows by expre ssing u in terms of T as in (3.3 3). ⊓ ⊔ 18 Krzysztof D¸ ebic ki et al. 4 App endix Let X = ( X 1 , . . . , X d ) a nd Y = ( Y 1 , . . . , Y d ) b e tw o Gaus sian random vectors with N (0 , 1) comp onents and co v aria nce matrices Σ 1 = ( σ (1) ij ) and Σ 0 = ( σ (0) ij ), resp ectively . The most elab orated version of Berman’s inequality is due to Li and Sha o [22], where it is shown that for u = ( u 1 , . . . , u d ) ∈ R d (hereafter the notation x ≤ y for a ny x , y ∈ R d means x i ≤ y i for all i ≤ d )     P ( X ≤ u ) − P ( Y ≤ u )     ≤ 1 2 π X 1 ≤ i 0     P ( X n : n ≤ u ) − P ( Y n : n ≤ u )     ≤ n (2 π ) n u − 2( n − 1) X 1 ≤ i − u s } ∩ n t =2 { Z h it ≤ − u i , Z h lt ≤ − u l }  , where W h s = max 1 ≤ j ≤ n Z h sj , a nd ϕ ( − u i , − u l ; τ h il ) is the biv ariate pdf of ( Z h i 1 , Z h l 1 ) which satisfies ϕ ( − u i , − u l ; τ h il ) = 1 2 π (1 − ( τ h il ) 2 ) 1 / 2 exp  − u 2 i − 2 τ h il u i u l + u 2 l 2(1 − ( τ h il ) 2 )  ≤ 1 2 π (1 − ( τ h il ) 2 ) 1 / 2 exp  − u 2 1 + ρ il  , u = min 1 ≤ i ≤ n u i . Next, let ( Z i , ˜ Z l ) b e a biv ar iate standa rdized normal r andom vector with cor- relation | τ h il | and set u = min 1 ≤ i ≤ n u i > 0. Sle pia n’s inequality in [25] and Lemma 2.3 in [24] imply P  Z h it ≤ − u i , Z h lt ≤ − u l  ≤ P  Z i ≤ − u i , ˜ Z l ≤ − u l  ≤ (1 + | τ h il | ) 2 u 2 ϕ ( u, u ; | τ h il | ) , t ≤ n. 20 Krzysztof D¸ ebic ki et al. Consequently , with A ∗ il defined by (4.36) and x + = max( x, 0) P ( X n : n ≤ u ) − P ( Y n : n ≤ u ) ≤ n (2 π ) n u − 2( n − 1) X 1 ≤ i

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