An Extension of Almost Sure Central Limit Theorem for Order Statistics

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1 An Extension of Amost Sure Centra Limit Theorem for Order Statistics T. Bin, P. Zuoxiang & S. Nadarajah First version: 6 December 2007 Research Report No. 9, 2007, Probabiity Statistics Group Schoo of Mathematics, The University of Manchester

2 An Extension of Amost Sure Centra Limit Theorem for Order Statistics a Tong Bin a Peng Zuoxiang b Saraees Nadarajah a Schoo of Mathematics Statistics, Southwest University, Chongqing, 40075, China b Schoo of Mathematics, University of Manchester, Manchester, United Kingdom Abstract. Let {X n, n } be a sequence of iid rom variabes. Let M n () M n (2) denote the first the second argest maxima. Assume that there are normaizing sequences a n > 0, b n a nondegenerate imit distribution G, such that a n (M n () b n ) d G. Assume aso that {d, } are positive weights obeying some mid conditions. Then for x > y we have im = d I{ M () b x, M (2) a a b y} = G(y){og G(x) og G(y)+} a.s. when G(y) > 0 ( to zero when G(y) = 0). AMS Cassification. Primary 60F5; Secondary 60F05. Keywords. Max-stabe distribution; Summation methods; Amost sure centra imit theorems; Extreme order statistics. Introduction Starting with Brosamer (988) Schatte (988), severa authors have investigated the amost sure centra imit theorem for partia sums of independent rom variabes. The simpest form of the a.s. centra imit theorem states that if {X n, n } are iid rom variabes with E(X ) = 0, E(X) 2 =, then for the partia sums S n = n = X, we have im og n = I{ S x} = Φ(x) a.s., (.) where Φ( ) is the stard norma distribution function I{ } sts for the indicator function. The universa version of amost surey centra imit theorems considered by Beres Csái (200) incudes the case of the maximum of iid sequence (Fahrner Stadtmüer, 998; Cheng et a., 998). Let X, X 2,, X n denote the iid rom variabes with common distribution function F (x), M n = max i n X i. Assume F is in the domains

3 of attraction of G γ, γ R, the extreme vaue distribution, i.e. there exist normaizing sequences a n > 0, b n such that a n (M n b n ) d G (.2) (see e.g. Leadbetter et a. (983) Resnic (987)). Fahrner Stadtmüer (998) Cheng et a. (998) proved that im og n = I{(a (M b ) x} = G(x) a.s. (.3) Note that with ordinary averages (.3) becomes fase even for x = 0 by the arc sine aw. Beres Csái (200) showed that if then im d = exp((og ) α )/, 0 α < /2 (.4) = d I{ M b a x} = G(x) a.s. (.5) As (.4) gives ogarithmic averaging for α = 0 ordinary averaging for α =, reation (.5) means that we can go at east hafway from ogarithmic to ordinary averaging. Recenty, Hörmann (2007) extended this resut showed that ogarithmic means, traditionay used in a.s. centra imit theory, can be repaced by other means ying much coser to ordinary averages. The anaogous resuts for partia sums was treated by Hörmann (2006). In this paper, we are interested in the reated questions for order statistics. As before, et {X n, n } denote a sequence of iid rom variabes et X,n X 2,n X n,n denote the order statistics of {X,..., X n }. The asymptotic distribution of the first the second argest maxima was obtained by Leadbetter et a. (983). Let M n () denote the first the second argest maximum of X,..., X n. Suppose that M (2) n for some nondegenerate d.f. G. Then, for x > y, P (a n (M () n b n ) x) d G(x) (.6) P (a n (M n () b n ) x, a n (M n (2) b n ) y) d G(y){og G(x) og G(y) + } (.7) when G(y) > 0 ( to zero when G(y) = 0) (cf. Leadbetter et a. (983)). The purpose of this paper is to obtain an amost sure version reated to (.6) (.7). 2 Main Resuts Theorem 2.. Let {X n, n } be a sequence of iid rom variabes. Let M n () M n (2) denote the first the second argest maximum of X,..., X n. Assume that there are normaizing sequences a n > 0, b n a nondegenerate imit distribution G (a so-caed max-stabe distribution) such that a n (M () n b n ) d G. (2.) 2

4 Assume aso that d, are positive weights satisfying d, (2.2) d is eventuay nonincreasing (2.3) d where D = i d. Then for x > y we have D (og D ) ρ for some ρ > 0, (2.4) im = d I{ M () b x, M (2) a a b when G(y) > 0 ( to zero when G(y) = 0). y} = G(y){og G(x) og G(y) + } a.s. (2.5) Theorem 2.2. Let {X n, n } be a sequence of iid rom variabes. Assume aso that f(x, y) is a bounded Lipschitz function. Then under the condition of Theorem 2., we have im d f( M () b, M (2) b ) = f(x, y)dg(x, y) a.s., (2.6) a a = where G(x, y) = G(y){og G(x) og G(y) + }. Theorem 2.3. Let {X n, n } be a sequence of iid rom variabes. Assume aso that g(x) is a bounded Lipschitz function. Then under the condition of Theorem 2., we have x>y im = d g( M () b ) = a g(x)dg(x) a.s. (2.7) Remar 2.. Let = exp((og n) α ) for some 0 < α <. Then d α exp((og )α ) (og ) α so (2.2) hods for d,. For d α exp((og )α ), et f(x) = exp((og x)α ). We get (og ) α (og x) α f (x) = exp((og x)α )(2/x ( α) (og x) α ) (og x) 2 2α < 0 for arge x. Thus, d is eventuay nonincreasing (2.3) hods. As d exp((og )α ) (og ) α = D (og ) α D (og D ) α we see (2.4) hods with ρ = α. Thus, Theorems show that ogarithmic means can be repaced by other means much coser to ordinary averages, eading to consideraby sharper resuts. If d = / the foowing resut hods. 3

5 Coroary 2.. Let {X n, n } be a sequence of iid rom variabes. Assume aso that there are normaizing sequences a n > 0, b n a max-stabe distribution G such that Then, (i). im og n (ii). For x > y, im og n P (a n (M n () b n ) x) d G(x). I{a = I{a = () (M () (M = G(y){og G(x) og G(y) + } a.s. when G(y) > 0 ( to zero when G(y) = 0). b ) x)} = G(x) a.s. b ) x, a (2) (M b ) y} 3 Proofs Lemma 3.. Let Z, Z 2, Z p be a sequence of rom variabes, then E Z Z 2 Z p (E Z p ) p (E Z2 p ) p (E Zp p ) p. (3.) Lemma 3.2. Let {X n, n } be a sequence of iid rom variabes with X F (x). Let M (2) M (2) denote, respectivey, the second argest maximum of (X,..., X ) (X +,..., X ). Then { P (M (2) M (2) 3 2 ) < 2. Proof. It is cear that P (M (2) M (2) ) = when < 2. For 2, we cacuate the probabiity of the compement. Note that the sequence {X n, n } is iid. We have P (M (2) = M (2) ) = ( )( )P (X > X > max {X v} > max {X j}) + v 2 j = ( )( ) = τ τ2 t ( )( )( 2). ( 2)( ) df (s) df (t) 2 df (τ 2 )df (τ ) 4

6 Thus, we obtain P (M (2) M (2) ( )( )( 2) ) = ( 2)( ) = ( )( )( 2 ) = exp{og( ) + og( ) + og( exp{3 og( 2 )} 3 og( 2 ) )} by observing that e x x, x > 0 og( x) x, 0 x. For n, denote η = I{M () u, M (2) v } P (M () u, M (2) v ), η = I{M () u, M (2) v } P (M () u, M (2) v ) (3.2) where u = a x + b v = a y + b. Q() = Q(, ) = η η, Lemma 3.3. Let {X n, n } be a sequence of iid rom variabes with X F (x). We have cov(η, η ) for n. (3.3) Proof. By the independence of M M, we have cov(η, η ) = cov(i{m () u, M (2) v }, I{M () u, M (2) v }) cov(i{m () u, M (2) v }, I{M () u, M (2) v } I{M () u, M (2) v }) + cov(i{m () u, M (2) v }, I{M () u, M (2) v } I{M () u, M (2) v }) + cov(i{m () u, M (2) v }, I{M () u, M (2) v }) = cov(i{m () u, M (2) v }, I{M () u, M (2) v } I{M () u, M (2) v }) + cov(i{m () u, M (2) v }, I{M () u, M (2) v } I{M () u, M (2) v }) := D + D 2. Now for D, D E I{M () u, M (2) v } I{M () u, M (2) v E I{M (2) v } I{M (2) v } P (M (2) M (2) ) = P (M (2) > M (2) ) { 3 2 < 2 5

7 by Lemma 3.2. Noting D 2 E I{M () u, M (2) v } I{M () u, M (2) v } = E I{M (2) v } I{M () u } I{M () u } E I{M () u } I{M () u } P (M () > M () ) = P (M () > M () ) = F (x) df (x) competes the proof of the Lemma. Lemma 3.4. Let {X n, n } be a sequence of iid rom variabes with η, η, Q() defined by (3.2). For any positive weights {d, }, we have for any m n p N E d Q() p E p ( d 2 ) p/2, (3.4) =m where E p = c 4 p p p/2 c is a constant. =m Proof. First we need the foowing moment inequaity: E Q() p = E η η p = E η η p η η 4 p E η η 2 4 p E I{M () u, M (2) v } I{M () u, M (2) v } 2 4 p E I{M () u, M (2) v } I{M () u, M (2) v } p E I{M () u, M (2) v } I{M () u, M (2) v } 2 4 p (E I{M (2) v } I{M (2) v } + E I{M () u } I{M () u } ) 2 4 p [P (M (2) > M (2) ) + P (M () > M () )] 4 p + 2 4p F (x) df (x) c 4 p. 6

8 By Lemma 3., we have E d Q() p = E =m = E d Q( ) =m =m =m =m d p Q( p ) p=m d d p Q( ) Q( p ) p=m d d p E Q( ) Q( p ) p=m d d p (E Q( ) p E Q( p ) p ) /p p=m c 4 p =m = c 4 p ( c 4 p m(( =m p=m d p ) p =m d d p ( p ) /p d 2 ) 2 ( 2 p ) 2 ) p =m =m = c 4 p ( d 2 ) p/2 m( 2 p ) p/2, where the ast inequaity foows by Cauchy-Schwartz inequaity. For m 2, For m = p 2, m( ( =m =m 2 p ) p/2 m( 2 p d) p/2 m 2 p ) p/2 = ( + = = m( p 2 (m ) 2 p ) p/2 = ( p 2 )p/2 m m p p/2. ( + 2 p ) p/2 =2 2 = ( + p 2 )p/2 p p/2. 2 p d) p/2 The proof is compete. 7

9 Lemma 3.5. Let {X n, n } be a sequence of iid rom variabes. Let {d, } be positive weights satisfying (2.2),(2.3),(2.4) with D = i d i. Then, for every p N, we have E d η p C p ( d d )p/2, (3.5) = n where C p = (γ) p/2, γ is chosen arge enough. Proof. Set Then V m,n = d ( d ), m n. =m V,n = = n We show that if γ is chosen arge enough then E d d. d η p C p (V m,n ) p/2 for a m n (3.6) =m this wi proof Lemma 3.5. We use induction on p to prove (3.6). For p = 2, using Lemma 3.3, we obtain E d η 2 2 =m = 2 m n m n m n C 2 m n d d Eη η d d cov(η, η ) d d d d if we choose γ arge enough. Thus, (3.6) hods for p = 2. For p =, by the Cauchy- Schwartz inequaity, we have E d η (E =m d η 2 ) /2 ( =m m n d d )/2 C ( m n d d )/2 if we choose γ arge enough. Suppose that (3.6) hods for p 2, then we sha prove it aso hods for p. We prove it in two parts. First, we show (3.6) hods for V mn γ. Notice that d. So, there exist some A > 0 such that d A. = 8

10 Thus, we have d ( d ) A =m = E d η p 2 p ( d ) p =m =m 2 p A p ( =m d d ( d )) p =m = = 2 p A p V p m,n 2 p A p γ p/2 V p/2 m,n C p V p/2 m,n if we choose γ arge enough such that (γ p2 ) 2 p A p γ p/2. We now prove (3.6) aso hods for V m,n > γ. More precisey, we show that if H γ is arbitrary (3.6) hods for V mn H, then it wi aso hod for V mn 3H/2. Assume that V mn 3H/2, set choose w such that Set aso d η = =m w d η + =m =w+ d η := G + G 2 V m,w H, V w+,n H, V w+,n V m,w = λ [/2, ]. T 2 = =w+ d η w,. By the binomia formua the triange inequaity, we obtain E p d η p E G p + E G 2 p + Cp(E G j j G p j 2 T p j 2 + E G j T 2 p j ) =m j= p = E G p + E G 2 p + Cp(E G j j G p j 2 T p j 2 + E G j E T 2 p j ) j= p E G p + E G 2 p + Cp[(p j j)e G j G 2 T 2 ( G p j 2 + T p j 2 ) j= + E G j E T 2 p j ], (3.7) where the equaity foows from the independence of G 2 T 2 the second inequaity foows from the mean vaue theorem. By hypotheses, we get E G p C p V p/2 m,w, (3.8) 9

11 E G 2 p C p λ p/2 V p/2 m,w (3.9) E G j C j V j/2 m,w for j p. (3.0) Notice that d is eventuay nonincreaing, so there exist some B < 0 such that By Lemma 3.4, we obtain B d d 2. = E G 2 T 2 j = E E j ( = E j ( =w+ =w+ =w+ E j B j/2 ( d (η η w, ) j d 2 ) j/2 d d 2 ) j/2 =w+ = E j B j/2 V j/2 w+,n = F j λ j/2 V j/2 m,w, d where F j = E j B j/2. Hence, by Minowsi s inequaity, we get E T 2 j [(E G 2 T 2 j ) /j + (E G 2 j ) /j ] j d ) j/2 = C 0 j E G 2 j + C j (E G 2 T 2 j ) /j (E G 2 j ) (j )/j + + C j j E G 2 T 2 j (Cj 0 + Cj + + C j j )C jλ j/2 Vm,w j/2 = 2 j C j λ j/2 Vm,w. j/2 (3.) Using Lemma 3., we obtain E G j G 2 T 2 G 2 p j (E G p ) j/p (E G 2 T 2 p ) /p (E G 2 p ) (p j )/p Simiary, we get = (C p V p/2 m,w) j/p (F p λ p/2 V p/2 m,w) /p (C p λ p/2 V p/2 m,w) (p j )/p = (C p ) (p )/p (F p ) /p λ (p j/)/2 (V m,w ) p/2. (3.2) E G j G 2 T 2 T 2 p j 2 p j (C p ) (p )/p (F p ) /p λ (p j/)/2 (V m,w ) p/2. (3.3) 0

12 By (3.8)-(3.3), we have for (3.7) E G + G 2 p C p (V m,w ) p/2 + C p λ p/2 (V m,w ) p/2 Notice that + + p Cp(p j j)( + 2 p j )(C p ) (p )/p (F p ) /p λ (p j/)/2 (V m,w ) p/2 j= p CpC j j Vm,w(2 j/2 p j C p j λ (p j)/2 V m,w (p j)/2 ) j= C p (V m,w ) p/2 ( + λ p/2 ) + C p (V m,w ) p/2 [(C p ) /p F /p p + p j= p 2 p j Cp(p j j)λ (p j)/2 j= C j C p j C p C j p2 p j λ (p j)/2 ]. (C p ) /p (F p ) /p = (γ p2 ) /p (c 4 p p p/2 B p/2 ) /p c 2 γ p p /2 (3.4) Observing that λ, we get C j C p j C p = γj2 γ (p j)2 γ p. (3.5) γ p2 p p p 2 p j Cp(p j j)λ (p j)/2 p 2 p j Cp j p 2 p j Cp j = p 3 p (3.6) j= j= j= So, (3.4)-(3.7) impy p Cp2 j p j λ (p j)/2 j= p 2 p j Cp j = 3 p. (3.7) j= (C p ) /p F /p p p 2 p j Cp(p j j)λ (p j)/2 c p 3/2 3 p γ p 0 as γ j= we obtain p j= C j C p j C p C j p2 p j λ (p j)/2 γ p 3 p 0 as γ E G + G 2 p C p (V m,w ) p/2 ( + λ p/2 ) C p (V m,w ) p/2 ( + λ) p/2 = C p (V m,w ) p/2 when γ is chosen arge enough. Hence, we have competed the proof of Lemma 3.5.

13 Lemma 3.6. Assume that {d, } are positive weights satisfying (2.4). We have for some 0 < η < ρ. Proof. Note n For I, we get d d = For I 2, using (2.4), we get I 2 n n (og D N ) ρ = where η < ρ. The proof is compete. = = D 2 n d d = O( (og ) ) (3.8) η d d + I d (og ) ρ (og ) ρ n >(og D N ) ρ D 2 N (og ) ρ. (og ) ρ = d D (og D ) ρ (og ) ρ d og og = (og ) og og D ρ n (og ), η d d := I + I 2. Proof of Theorem 2.. Denote µ n = n = d η. By Lemma 3.5 Lemma 3.6 the Marov inequaity, we derive for any ε > 0, p N P ( = d η > ε) E n = d η p ε p ε C p( p n d d )p/2 Dn p C p ε p Dp n(og ) pη/2 Dn p = C(p, ε)(og ) pη/2. Seect n j = inf{n : exp( j)} for every j. We get j exp( j) j + j exp( j + j) as j. 2

14 Hence, P ( j n j d η > ε) C(p, ε)j pη/4 < = if we choose some p such that p > 4/η, then, by Bore-Cantei Lemma, we get µ nj = j n j d η 0 = a.s. Let be such that n < n n +. We have µ n = d η j = n j = d η + n j = d η + j = µ nj + 2( j+ j ) so µ n 0 a.s.. The proof is compete. n j+ =n j + n j+ =n j + d η Proof of Theorem 2.2 Theorem 2.3. The proofs are simiar to that of Theorem 2.. 2d References [] Brosamer, G. (988). An amost everywhere centra imit theorem. Mathematica Proceedings of the Cambridge Phiosophica Society, 04, [2] Beres, I. Csái, E. (200). A universa resut in amost sure centra imit theory. Stochastic Processes Their Appications, 9, [3] Cheng, S., Peng, L. Qi, Y. (998). Amost sure convergence in extreme vaue theory. Mathematische Nachrichten, 90, [4] Fahrner, I. Stadtmuer, U. (998). On amost sure max-imit theorems. Statistics Probabiity Letters, 37, [5] Hömann, S. (2006). An extension of amost sure centra imit theory. Statistics Probabiity Letters, 76, [6] Hömann, S. (2007). On the universa a.s. centra imit theorem. Acta Mathematica Hungarica, 6,

15 [7] Leadbetter, M. R., Lindgren, G. Rootzen, H. (983). Extremes Reated Properties of Rom Sequences Processes. Springer, Berin. [8] Resnic, S. I. (987). Extreme Vaues, Reguar Variation Point Processes. Springer, Berin. [9] Schatte, P. (988). On strong versions of the centra imit theorem. Mathematische Nachrichten, 37,