理学院毕业（设计）论文 第 31 页
原文1：
Almost Sure Versions of Central Limit Theorems for Order
Statistics
Abstract: Let{Xn}be a sequence of i.i.d random variables with distribution function
F(x)，Mn(1)、Mn(2)denote, respectively, the first and the second largest maximum of
{X1,X2,...,Xn,}，assume also that there are normalizing sequences an?0,bnand a
non-degenerate limit distribution G(x) such that
P(Mn?anx?bn)???G(x)(1)w,then for x>y we have an almost sure central limit
theorem for Mn(1) and Mn(2),i·e·
lim1logNNN???n?11nI?Mn?un,Mn(1)(2)?vn??G(y){logG(x)?logG(y)?1}
a.s.
where un?anx?bn,vn?any?bn.
Key words: almost sure central limit theorem; non-degenerate limit distribution; extreme order statistics
4 Introduction
Initiated by the papers of [1] and [2], many authors have discussed almost sure versions of central limit theorems, which states for independent identically distributed (i·i·d·) random variables{Xn}withE?X1??0,E?X12??1 for the partial sums
nSn??k?1Xkwe have
1logNN1?1??2I?nSn?x???(x) a.s. (1) n?? limN???n?1where?(x)is the standard normal distribution, IAstands for the indicator function. The universal version of almost surely central limit theorems considered by [3]
includes the case of the maximum of i·i·d· sequence, which was first studied by [4] and [5] respectively. Let us denote Mn?maxXiand assume that there exist normalizing
1?i?nsequencesan?0,bnand a non-degenerate limit distribution G(a so-called max-stable distribution), such that
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P(an(Mn?bn)?x)???G(x) (2)
?1(1)w(ee, e·g·[6] and [7]), [4] and [5] proved the following result respectively
lim1logNNN???n?11nI?an(Mn?bn)?x??G(x) a.s. (3)
For dependent case, [8] obtained almost sure max-limit theorems of stationary Gaussian
sequence provided its covariance satisfying some regular conditions, later [9] extended the related results to the case of stationary Gaussian vector sequence under some mild conditions on covariance.
Recently [10] studied the almost sure central limit theorem for intermediate order statistics and central order statistics. We are also interested here in the related questions for order statistcs. As before?Xn?is a sequence of i·i·d·random variables with associated order statisticsX1,n?X2,n?...?Xn,n.The joint asymptotic distribution of the first and second largest maximum was obtained by Leadbetter et al. (see[6]), which is the following theorem.
Theorem A LetMn(1)?Xn,n,Mn(2)?Xn?1,ndenote, respectively, the first and the second largest maximum of?X1,X2,...,Xn?and suppose that (2) holds, then for x>y,
P(an?1(Mn?(1)b)?nx,an?1(M?n(2)?b)n?y)??wG)log?G(x)?(ylo?Gg? y( （)4）1
when G(y) >0 (and to zero if G(y) =0).
The purpose of this paper is to obtain an almost sure version of Theorem 2·3·1 of [6]. Since the mere existence of the weak limit does not always imply the almost sure limiting result for the logarithmic averages (see [3]), the investigation of a general almost sure limit theorem seems somewhat challenging.
5 Main results
In this section we are interested in the almost sure central limit theorems of random vector(Mn(1),Mn(2)).The main results are provided here and all proof are deferred to Section 3. For convenience, in the rest of the paper, we denote
un?un(x)?anx?bn,vn?vn(y)?any?bn
Theorem 1 Let ?Xn?be a sequence of i·i·d· random variables, Mn(1)，Mn(2)denote, respectively, the first and the second largest maximum, and assume (2) hold, then for x > y
N??lim1logNN?nI{an?11?1n(Mn?bn)?x,an(Mn?bn)?y}?G(y)?logG(x)?logG(y)?1? a.s.
(1)?1(2) 理学院毕业（设计）论文 第 33 页
(5)
as G(y) >0 (and to zero if G(y) =0).
Based on Theorem 1, we derived the well-known almost sure central limit theorem of the maximum of the original i·i·d· random variables {Xn}, which, as we have mentioned, was firstly provided by [4] and [5].
Corollary 1 Under the conditions of Theorem 1, for all real x we have
N??lim1logNN?n?11nI?an(Mn?1(1)?bn)?x??G(x) a.s. (6)
Following corollary extended the desired result to bounded Lipschitz function case. corollary 2 Let{Xn}be a sequence of i·i·d· random variables, assume also that f(x, y)is a bounded Lipschitz 1 function, Then under the condition of Theorem 1, we have
N??lim1logNN?n?11nf(an(M?1(1)n?bn),an(M?1(2)n??bn))???f(x,y)dG(x,y) a.s. (7)
where G(x,y)?G(y){logG(x)?logG(y)?1}
6 Proofs
We need some technique to prove our main results. Following lemma will be crucial in the whole proof of our main theorem.
?Lemma 1 Let ??n?be a sequence of bounded random variables. If
NVar(?n?11n?n)?(logN)2??， (8)
Then
NVar(?n?11n(?n?E?n)?0a.s. (9)
Proof See Lemma 1 in [5].
(2)Lemma 2 Let{Xn}be a sequence of i·i·d· random variables, andMn(2),Mmdenote, ,nrespectively, the second largest maximum of{X1,...Xm}and{Xm?1,...Xn}then
m?2?(2)?Mm,n)??n?1,n?m?2,m?1
?1,其它?P(Mn(2)(2))?1 when n-m<2.Forn-m≥2,we calculate the Proof It is clear that P(Mn(2)?Mm,nprobability of the complement. Note that the sequence ?Xn?is i·i·d·, we have
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P(Mn(2)?Mm,n)?(n?m)P(Xn???1?2(2)m?1?v?n?1max{Xv}?max{Xj})1?j?m?(n?m)????dF(t)dF(?2)mn?1?mdF(?1)
??????(n?m)(n?m?1)n(n?1)Therefore, we obtain
P(Mn(2)?Mm,n)?1?mn)(1?2(2)(n?m)(n?m?1)n(n?1)?1?(1??1?(1?nmn?1)m
mn?1ni)?2n?1nby the elementary fact that
?x??i?1i?1yi??i?1xi?yihold for xi?1,yi?1.The result
follows. Let us denote?n?I{Mn(1)?un,Mn(2)?vn}?P{Mn(1)?un,Mn(2)?vn} Following pivotal lemma is estimating the bound of the covariance of {?n} Lemma 3 Under the conditions of Theorem 1, for n-m?2 we have
Cov(?m,?n)?mn?1
Proof By elementary calculations, for m < n we have
Cov(?m,?n)?Cov(I{Mm?um,Mm?vm},I{Mn?un,Mn(1)(2)(1)(2)?vn})
?EI{Mn?un,MnEI{M(1)n(1)(2)?vn}?I{Mn?un,Mm,n?vn}?(1)m,n(1)(2) (10)
?un,M(1)(2)m,n?vn}?I{M(2)?un,M(1)(2)m,n?vn}?(2)Cov(I{Mm?um,Mm?vm},I{Mm,n?um,Mm,n?vm})?A1?A2?A3Note that A3=0 by independence of?Xn?,our task now is to estimate the upper bound ofA1 and A2 in (10).
By the same statement of [3], p. 123, for1?m?n we have
理学院毕业（设计）论文 第 35 页 A2?EI{Mn?un}?I{Mm(1)(1)(1)(1)n,?un}IM{mn,?vn(2)} ?EI{Mn?un}?I{Mm,n?un}?P(Mn?Mm,n)?(1)(1) （11）
mnNow it remains to estimate the bound of component A1,By using Lemma 2, for n-m?2, we have
A1?EI{Mn(2)（2）(2)?un}?I{Mm,n?un}?P(Mm?Mm,n)?P(Mn(2)(2) (12)
mn?1?Mm,n)?2(2)Relations (10), (11), (12) yield the desired result.
Proof of Theorem 1 By Lemma 1, we only need to estimate the bound of weighted sum of??n?.Notice that
12Var(??n)?E(??n)n?1nn?1nNN1N??n?11nE?n?222?1?m?n?N1mnE?m?n (13)
?B1?B2Since?n?2,we have
?B1??n?11n2?? (14)
Now we start to estimate component B2.By Lemma 3, we have
B2??1?m?n?Nn?m?21mn??1?m?n?Nn?m?21mmnn?1
?logN (15)
Now relations (13)-(15), combined with Lemma 1, imply the desired result.
Proof of Corollary 1 Observing that for any ??0the following inequalities hold:
I{Mn?anx?bn}?I{Mn?an(x??)?bn,MnI{M(1)n(1)(1)(2)?anx?bn}?anx?bn}?I{M(1)n?anx?bn,M(2)n?an(x??)?bn}
Therefore, by using Theorem 1, we have