Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 856495,14 pages doi:10.1155/2010/856495

Research Article

Almost Sure Convergence for the Maximum and the Sum of Nonstationary Guassian Sequences

Shengli Zhao,1 Zuoxiang Peng,2 and Songlin Wu1

1 Department of Fundament Studies, Logistical Engineering University, Chongqing 401131, China

2 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Correspondence should be addressed to Shengli Zhao, zhaoshengli83@yahoo.com.cn Received 18 December 2009; Accepted 5 April 2010 Academic Editor: Jewgeni Dshalalow

Copyright © 2010 Shengli Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let (Xn, n > 1) be a standardized nonstationary Gaussian sequence. Let Mn = max{Xk, 1 < k < n} denote the partial maximum and Sn = ^n~ 1 Xk for the partial sum with an = (Var Sn)1/2. In this paper, the almost sure convergence of (Mn, Sn/an) is derived under some mild conditions.

1. Introduction

There have been more researches on the almost sure convergence of extremes and partial sums since the pioneer work of Fahrner and Stadtmuller [1] and Cheng et al. [2]. For more related work on almost sure convergence of extremes and partial sums, see Berkes and Csaki [3], Peng et al. [4, 5], Tan and Peng [6], and references therein. For the almost sure convergence of extremes for dependent Gaussian sequence, Csaki and Gonchigdanzan [7] and Lin [8] proved

= exp(-e-x-p+v^z)0(z)dz

provided

Tn log n - p| (log log n)1+£ = 0(1),

where I denotes an indicator function, ®(x) is the standard normal distribution function, and $(x) = (1/V2n)e-x2/2 = ®'(x). Mn is the partial maximum of a standard stationary Gaussian

sequence {Xn, n > 1} with correlation rn = EXiX„+i, n > 0. The norming constants an and bn are defined by

a. = <2log„,-1'2, b„ = (2logn)i/2 - 'og2*og" + 'og4j (1.3)

2(2logn) '

For some extensions of (1.1), see Chen and Lin [9] and Peng and Nadarajah [10].

Sometimes, in practice, one would like to know how partial sums and maxima behave simultaneously in the limit; see Anderson and Turkman [11] for a discussion of an application involving extreme wind gusts and average wind speeds. Peng et al. [12] studied the almost sure limiting behavior for partial sums and maxima of i.i.d. random variables. Dudzinski [13, 14] proved the almost sure limit theorems in the joint version for the maxima and the partial sums of stationary Gaussian sequences, that is, let X1/X1/... be stationary Gaussian sequences and Mk = maxi<kXi, Sn = ^n=1 Xi, on = -\/Var(Sn), for all x,y e (-to, to)

lim-^- Vh(Mk - bk < xS < ^ = exp(-e-x)0(y) a.s. (1.4)

n —<x,log n k= k \ ak Ok V ^ y

(C1) sups>nlft=-ln \rt\ « (log n)1/2/(log log n)1+£ for some £ > 0, (C2) En=1(n - t)rt > 0 for all n > 1, (C3) lim n —> to rn log n = 0.

rn = ^^, n > 1 (1.5)

for some a > 0. L(x) is a positive slowly varying function at infinity. Here a « b means a = O(b).

This paper focuses on extending (1.4) to nonstationary Gaussian sequences {Xn, n > 1} under some mild conditions similar to (C1)-(C3). The paper is organized as follows: in Section 2, we give the main results, and related proofs are provided in Section 3.

2. The Main Results

Let rij = E(XiXj), i,j > 1, denote the correlations of standard nonstationary Gaussian sequence {Xn, n > 1}. Mn, Sn, and on are defined as before. The main results are the following.

Theorem 2.1. Let {Xn, n > 1} be a standardized nonstationary Gaussian sequence. Suppose that there exists numerical sequence {uni, 1 < i < n, n > 1} such that ^n=1 (1 - ®(uni)) — T for some 0 <t < to and n(1 - 0(Xn)) is bounded, where Xn = min1<i<nuni. If

sup|'rijI <6< 1, (21)

i/j K ' '

EElrij I = °(n)

j=2 ¿=1

Tii «

(tog n)

¿>1 j=1 ^ (log log n)1+£

for some e> 0,

(2.2) (2.3)

1 n i / k

lim i—E zHf) (xi < uki), — < y =®(y) i^œlognk=1k \U °k '

for all y e (-to, to).

Theorem 2.2. For the nonstationary Gaussian sequence {Xn, n > 1}, under the conditions (2.1)-(2.3), we have

1 n 1 / S \

lim --Y-I( Mk < akx + bk,— < y ) = exp(-e~x)y) a.s. (2.5)

t^^lognk=1 k \ Ok V

for all x,y e (-to, to), where a„ and bn are defined as in (1.3).

3. Proof of the Main Results

To prove the main results, we need some auxiliary lemmas.

Lemma 3.1. Suppose that the standardized nonstationary Gaussian sequences {Xn, n > 1} satisfy the conditions (2.1)-(2.3). Assume that n(1 - 0(Xn)) is bounded. Then for < l,

l( D (Xi < uu),S < y) -1( H (Xi < uii),S < y ¿=1 Ol / \i=k+1 Ol

(log logl)

1+£ l

Proof. We will start with the following observations. For all 1 < i < l,

Ccv(Xi,S) = -1Cov(Xi,Sl)| < -¿|

Clearly,

Ol = (l +

j=2 ¿=1

By (2.2), for large l there exists c1 > 0 such that

Oi > C1l1/2.

By (2.3) and (3.4), we have

1< i<l

Covf XUS

(log l)

l1/2(log log l)

for large l. Obviously,

lim (l°g 1)1/2 1 . 0,

l—l1/2(log log l)1+£

which implies that there exist fi> 0 and l0 such that

Notice,

Co< X.S)

< p< 1 Vl > l0.

I H (Xi < Uli),sL < y - fl ft (Xi < un), -— < y

= P( f| (Xi < Uli), S < y) - P(f| (Xi < Uli),S < y

P(Q (Xi < Uli), S < yj - Q (Xi < Uli) J p(S < y)

P( f (Xi < Un),- < y) - P( ft (Xi < Un)) p( S < y \i=fc+1 Ol / \i=fc+1 /

+ p(S < y) ^ ff++ (Xi < Uli)^ - P^f (Xi < Uli)

=: A (l)+ A2(l)+ A3(l).

Covf Xi,Sl

22 K + y2

By the Normal Comparison Lemma [13, Theorem 4.2.1 ], we get

A1(l) «X

2(1 + | Cov(Xi,Sl/ol) |)

Covf XUS

2(1 + p)

Since n(1 - ®(ln)) is bounded, for large n and some absolute positive constant C,

exP< 4) ~ C^

(3.10)

A (l) « l1/2(log l)1/2 (log l)1/2(1+^) = (log l)1/2+1/2(1+^ ^ «

(log log l)1/2 l1/(1+^) l1/(1+^)-1/2(log log l)U£ (log log l)

(3.11)

Similarly,

A2(l) «

(log log l)

(3.12)

It remains to estimate A3(l). It is easy to check that

A3(l) < P( D (Xi < uu)\ - p(ff (Xi < uli)

¿=k+1 ¿=1

p( Q(Xi < uu)) - ol(Xl)

p( n (Xi < uli)) - Ol-k (Xl)

(W-k (Xl) - ol

=: B1(l)+ B2(l)+ B3(l).

(3.13)

By the arguments similar to that of Lemma 2.4 in Csaki and Gonchigdanzan [7], we get

B3(l) « j.

(3.14)

By the Normal Comparison Lemma and (3.4), we derive that

B1(l) « 1 expv 2(1 + D

ul2i + Xl2

< l l rijl exp -/ 1<i<l \

JL 1 + 6

(log l)1/2 (log l)1/(1+6) (log log l)1+e l2/(1+6) 1

(log log l)

(log log l)

(3.15)

Combining with above analysis, we have

* /TN k 1

A3(l) « 1 + n—. (3.16)

l (log log l)

The proof is complete. □

We also need the following auxiliary result.

Lemma 3.2. Suppose that the standardized nonstationary Gaussian sequences {Xn, n > 1} satisfy the conditions (2.1)—(2.3). Assume that n(1 - ®(ln)) is bounded; then

Cov( i( Q (Xi < un), S < y\if Q (Xi < un), S < y

for k < min(^2l(log log )2+27c22 log l,l), where 0 <$< 1, C2 > 0. Proof. By (2.2) and (2.3), for i>k + 1, we get

k1/2(log l)

«-v & ' , (3.17)

l1/2(log logl)1+£ V '

(log l)

l1/2(log log l)

1+£ '

(3.18)

Clearly,

(log l)

l1/2(log log l)

0 as l —> oo,

(3.19)

which implies that there exist g > 0 and k0 such that for k > k0,

ss(Cov( 1

(3.20)

For k < l, we have

OkOl Ok

O + Cov(Sk, Si) 1

< O + O^Z SK1

l k l i=1 j=i+1

Condition (2.2) implies that there exist positive numbers c3 and c4 such that c3k1/2 < ok < c4k1/2 and

( Si Si \

\ Ok , Ol J

k1/2 k1/2 (log l)1/2

« TTTT" +

l1/2 l1/2(log logl)1+e

k1/2(log l)1/2

l1/2(log logl)Ue'

(3.22)

So there exists 0 <v< 1 such that

<v < 1

(3.23)

for l1 < k < min(^2l(log log l) + £/c2logl,l).By applying the inequalities above and the Normal Comparison Lemma, we get

Cov(VD (Xi < uki),— <y\l( ff (Xi < uki),-<y

\ \i=i Ok J \,-=t+i Ol

(Sk S{ \

X1 < uk1,...,Xk < ukk,— < y,Xk+1 < ul(k+1),...,Xu < uji,-1 < y ) Ok Ol

-^X1 < uk1,...,Xk < ukk,-! < y)^Xk+1 < ul(k+1),...,Xll < un,- < y^

k l ( ui + u2

« S,J+1|r"1 exp(-;

Cov Xi,Sl

22 u2ki + y2

2(1 + |Cov(Xi, Si/oi )|)

Covf X,,-1

22 u2+y2

( Si Si \

\Ok' Ol)

2(1 + | Cov(Xj, Sk/Ok) |) 1\

1 + | Cov(Sk/ok, si/oi)|

=: D1(l)+ D2(l)+ D3(l)+ D4(l).

By (3.10), we have

Similarly,

D1(0 < Sjl^'H-2(1 + 6)

X2k +12

(log l)1/2 (log k)1/2(1+6) (log l)1/2(1+6) (log log l)1+£ k1/(1+6) l1/(1+6)

k1/2(log l)1/2

l1/2(log log l)

D2(l) < expl -

2(1 + P) j!k

Cov^ Xi,Sl

„ (log k)1/2(1+p) k (log l)

k1/(1+p) l1/2(log log l)1+ k1/2(log l)1/2

l1/2(log log l)

1+£ '

(3.25)

D3(l) < exp( -

\ 2(1 + g))j=k+1

\ J Ok

Cov Xj

< exp(-2(j)) S I11

(log l)1/2(1+g) (log l)1/2

l1/(1+9) k1/2 (log log l)1+£ k1/2

l1/2(log log l)

1+£ '

(3.26)

While (3.22) implies

D4(l) <

( si sI \

\Ok Ol)

k1/2(logl)

l1/2(log log l)

(3.27)

the proof is complete.

We also need the following auxiliary result.

Lemma 3.3. Let X1,X2,... be a standardized nonstationary Gaussian sequences satisfying assumptions (2.1)-(2.3). Assume that £ "=1(1 - 0(uni)) — t for some 0 <t < to and „(1 - ®(1„)) is bounded. Then

lim P ( ff (Xi < uki), — < y ) = e-o(y) (3.28)

k — to ^i=1 Ok J

for all e (-to, to).

Proof. By the Normal Comparison Lemma and the proof of Lemma 3.1, we have

p(q (Xi < Uki),Sr < y) - P^Q (Xi < Uki)^jp(^ < y)

«-^, (3.29)

(log log k)Ue

lim-= 0, (3 30)

k-to (log log k)Ue , ( )

which implies

lim p(f (Xi < ukl),— < y) = limP( f (Xi < uw) )p( — < y\ (3.31)

k-to yi=1 Ok J k-to yi=1 / \Ok /

By Theorem 6.1.3 of Leadbetter et al. [15], we have

limTOP(n (Xi < Uki= e~T. (3.32)

Since Sk/ok follows the standard normal distribution, we get

lim p(ff (Xi < Uki), — < y J = e-®(y), (3.33)

k -to yi=1 Ok J

which completes the proof. □

We now only give the proof of Theorem 2.1. Theorem 2.2 is a special case of Theorem 2.1.

Proof of Theorem 2.1. The idea of this proof is similar to that of Theorem 1.1 in Csaki and Gonchigdanzan [7]. In order to prove Theorem 2.1, it is enough to show that

VarfX 1 /ft (Xi < ukl), — < y)) « (log„)21 (3.34)

Vk=1k \U ( " Ok-")) (log log n)1+e ^ ;

for all fixed e (-to, to).

Let à = I(f||Li(Xi < Uki), Sk/ok < y) - P(f|h(Xi < m), Sk/ak < y), we have

n i / k

Var X k 1 H (Xi < uki),-r < y

k=1,v \i=1

n 1 1 <XT^lk + 2 Z 11 E(^)i=: Fl + F2.

k=1k 1<k<l<n

Since {¿,k} are bounded,

F1 « X fc2 < ^ k=1 k

The remainder is to estimate F2. Notice | E(£k£l )|= Cov

^jk (Xi < Uki), ^ < y), i( R (Xi < Uki), ^ < y

-1(0 (Xi < Uki),— < y

Vi=k+1 Ok

Cov( I(0 (Xi < Uki< y), l( f| (Xi < Uki)S < y

I H (Xi < Uki),Jk < y -1( 0 (Xi < Uki), — < y

i=i Ok / V=k+1 Ok

Cov( l( f| (Xi < Uki),— < y\II j (Xi < Uki),— < y

i=1 Ok 2+2e 1, 2 1

(3.35)

(3.36)

(3.37)

(3.38)

By Lemmas 3.1 and 3.2, we infer that if k< j32l(log log l) /(c2 logl) and k< 1,

|E(M)|« 1 , » k"2(log l)"l + k-(loglog n)1+e l,/2(log log l)1+£ l

for some e > 0. By the arguments similar to that of Theorem 1 in Dudzinski [13], we can get

(log n)2

(log log n)

So by Lemma 3.1 of Csaki and Gonchigdanzan [7] and Lemma 3.3,

(3.39)

1 n i / k

lim r^El HO (Xi < Uki), — < y ) = e-y) a.s.

logné1k VU Ok '

which completes the proof.

0.013 0.0125 0.012 0.0115

§ 0.011

5 0.0105

0.01 0.0095 0.009 0.0085

0 100 200 300 400 500 600 700 800

Figure 1: T he actual error, An, for rn = 1/[n(log n)1/2(log log n)] and (x, y) = (-1, -1).

0.04 0.038 0.036

§ 0.034

5 0.032

0.03 0.028 0.026

0 100 200 300 400 500 600 700 800

Figure 2: T he actual error, An, for rn = 1/[n(log n)1/2(log log n)] and (x,y) = (0,0).

4. Numerical Analysis

The aim of this section is to calculate the actual convergence rate of

^exp(-e-x)®(y)

1 n 1 / S

X1H Mk < fl-1x + bk,— < y

log nk=1k

for finite; that is, calculate

An(x,y) =

log nk=1k

n 1 / S \

XtH Mk < akx + bk, ^ < y ) - exp(-e~x)®(y) k=1k ^ Ok /

where an = (2logn) 1/2 and bn = (2logn)1/2 - (log log n + log4x)/2(2logn)1/2.

Figure 3: T he actual error, An, for rn = 1/[n(log n)1/2(log log n)] and (x, y) = (1,1).

0.055 -1-1-1-1-1-1-1-

0.05 -0.045 0.04 -o 0.035 -

s 0.03 -| 0.025 -0.02 -0.015

0.01 ; 0.005

0 100 200 300 400 500 600 700 800

Figure 4: T he actual error, An, for rn = 1/[n(log n)1/2 (log log n)(log log log n)] and (x, y) = (-1, -1).

Firstly, we will construct a standardized triangular Gaussian array {Xnj, 1 < j < n, n > 1} with equal correlation rn in n th array for > 1. Meanwhile, the sequence rn must satisfy the conditions (2.1), (2.2), and (2.3). By Leadbetter et al. [15], we can construct the Gaussian array by i.i.d Gaussian sequence; that is, let rn to a convex sequence, ... is a standardized i.i.d Gaussian sequence, and n is also a standardized normal random variable which is independent of ¿,k (k > 1). For each > 1, let

Xij = (1 - n)1^ + r1/2n, (4.3)

where = 1,2,...,i. Obviously, Xij (1 < j < i) is a zero mean normal sequence with equal correlation. By this way, we get the Gaussian array needed.

100 200 300 400 500 600 700

n 1/2/

Figure 5: T he actual error, An, for rn = 1/[n(log n) ' (log log n)(log log log n)] and (x, y) = (0,0)

0.25 0.2 g 0.15

0 100 200 300 400 500 600 700 800 n )1/2(

Figure 6: T he actual error, An, for rn = 1/[n(log n) ' (log log n)(log log log n)] and (x, y) = (1,1).

Figures 1 to 3 give the actual error, An, for rn = 1/[n(log n)1/2(log log n)] and (x,y) = (-1, -1), (0,0), (1,1). In each figure, the actual error shocks tend to zero as n increases. The overall performance of the actual error becomes better as (x, y) = (0,0). Figures 4 to 6 give the actual error, An, for

n(log n) 1/2 (log log n) (log log log n)j (x,y) = (-1,-1), (0,0), (1,1).

In each figure, the actual error shocks also tend to zero as n increases. Also the overall performance of the actual error becomes better as (x, y) = (0,0).

Acknowledgments

The authors wish to thank the referees for some useful comments. The research was partially supported by the National Natural Science Foundation of China (Grant no. 70371061), the

National Natural Science Foundation of China (Grant no. 10971227), and the Program for ET

of Chongqing Higher Education Institutions (Grant no. 120060-20600204).

References

[1] I. Fahrner and U. Stadtmuller, "On almost sure max-limit theorems," Statistics and Probability Letters, vol. 37, no. 3, pp. 229-236,1998.

[2] S. Cheng, L. Peng, and Y. Qi, "Almost sure convergence in extreme value theory," Mathematische Nachrichten, vol. 190, pp. 43-50,1998.

[3] I. Berkes and E. Csaki, "A universal result in almost sure central limit theory," Stochastic Processes and Their Applications, vol. 94, no. 1, pp. 105-134, 2001.

[4] Z. Peng, J. Li, and S. Nadarajah, "Almost sure convergence of extreme order statistics," Electronic Journal of Statistics, vol. 3, pp. 546-556, 2009.

[5] Z. Peng, L. Wang, and S. Nadarajah, "Almost sure central limit theorem for partial sums and maxima," Mathematische Nachrichten, vol. 282, no. 4, pp. 632-636, 2009.

[6] Z. Tan and Z. Peng, "Almost sure convergence for non-stationary random sequences," Statistics and Probability Letters, vol. 79, no. 7, pp. 857-863, 2009.

[7] E. Csaki and K. Gonchigdanzan, "Almost sure limit theorems for the maximum of stationary Gaussian sequences," Statistics and Probability Letters, vol. 58, no. 2, pp. 195-203, 2002.

[8] F. Lin, "Almost sure limit theorem for the maxima of strongly dependent gaussian sequences," Electronic Communications in Probability, vol. 14, pp. 224-231, 2009.

[9] S. Chen and Z. Lin, "Almost sure max-limits for nonstationary Gaussian sequence," Statistics and Probability Letters, vol. 76, no. 11, pp. 1175-1184, 2006.

[10] Z. Peng and S. Nadarajah, "Almost sure limit theorems for Gaussian sequences," Theory Probability and its Applidations. In press.

[11] C. W. Anderson and K. F. Turkman, "Limiting joint distributions of sums and maxima in a statistical context," Theory Probability and its Applidations, vol. 37, pp. 314-316,1993.

[12] Z. Peng, L. Wang, and S. Nadarajah, "Almost sure central limit theorem for partial sums and maxima," Mathematische Nachrichten, vol. 282, no. 4, pp. 632-636, 2009.

[13] M. Dudzirtski, "An almost sure limit theorem for the maxima and sums of stationary Gaussian sequences," Statistics and Probability Letters, vol. 23, pp. 139-152, 2003.

[14] M. Dudziriski, "The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences," Statistics and Probability Letters, vol. 78, no. 4, pp. 347-357, 2008.

[15] M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer, New York, NY, USA, 1993.

Copyright of Journal of Inequalities & Applications is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.