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

Abstract

Let (, ) be a standardized nonstationary Gaussian sequence. Let max denote the partial maximum and for the partial sum with (Var . In this paper, the almost sure convergence of (, ) 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 Stadtmüller [1] and Cheng et al. [2]. For more related work on almost sure convergence of extremes and partial sums, see Berkes and Csáki [3], Peng et al. [4, 5], Tan and Peng [6], and references therein. For the almost sure convergence of extremes for dependent Gaussian sequence, Csáki and Gonchigdanzan [7] and Lin [8] proved

(1.1)

provided

(1.2)

where denotes an indicator function, is the standard normal distribution function, and . is the partial maximum of a standard stationary Gaussian sequence with correlation . The norming constants and are defined by

(1.3)

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. Dudziński [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 be stationary Gaussian sequences and , , , for all

(1.4)

if

(C1)

(C2)

(C3)

Or

(1.5)

for some . is a positive slowly varying function at infinity. Here means .

This paper focuses on extending (1.4) to nonstationary Gaussian sequences 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 , denote the correlations of standard nonstationary Gaussian sequence . , and are defined as before. The main results are the following.

Theorem 2.1.

Let be a standardized nonstationary Gaussian sequence. Suppose that there exists numerical sequence such that for some and is bounded, where . If

(2.1)

(2.2)

(2.3)

then

(2.4)

for all .

Theorem 2.2.

For the nonstationary Gaussian sequence , under the conditions (2.1)–(2.3), we have

(2.5)

for all where and 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 satisfy the conditions (2.1)–(2.3). Assume that is bounded. Then for ,

(3.1)

Proof.

We will start with the following observations. For all ,

(3.2)

Clearly,

(3.3)

By (2.2), for large there exists such that

(3.4)

By (2.3) and (3.4), we have

(3.5)

for large . Obviously,

(3.6)

which implies that there exist and such that

(3.7)

Notice,

(3.8)

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

(3.9)

Since is bounded, for large and some absolute positive constant ,

(3.10)

So,

(3.11)

Similarly,

(3.12)

It remains to estimate . It is easy to check that

(3.13)

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

(3.14)

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

(3.15)

Combining with above analysis, we have

(3.16)

The proof is complete.

We also need the following auxiliary result.

Lemma 3.2.

Suppose that the standardized nonstationary Gaussian sequences satisfy the conditions (2.1)–(2.3). Assume that is bounded; then

(3.17)

for , where .

Proof.

By (2.2) and (2.3), for , we get

(3.18)

Clearly,

(3.19)

which implies that there exist and such that for ,

(3.20)

For , we have

(3.21)

Condition (2.2) implies that there exist positive numbers and such that and

(3.22)

So there exists such that

(3.23)

for .By applying the inequalities above and the Normal Comparison Lemma, we get

(3.24)

By (3.10), we have

(3.25)

Similarly,

(3.26)

While (3.22) implies

(3.27)

the proof is complete.

We also need the following auxiliary result.

Lemma 3.3.

Let be a standardized nonstationary Gaussian sequences satisfying assumptions (2.1)–(2.3). Assume that for some and is bounded. Then

(3.28)

for all .

Proof.

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

(3.29)

where

(3.30)

which implies

(3.31)

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

(3.32)

Since follows the standard normal distribution, we get

(3.33)

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 in Csáki and Gonchigdanzan [7]. In order to prove Theorem 2.1, it is enough to show that

(3.34)

for all fixed .

Let , we have

(3.35)

Since are bounded,

(3.36)

The remainder is to estimate . Notice

(3.37)

By Lemmas 3.1 and 3.2, we infer that if and ,

(3.38)

for some . By the arguments similar to that of Theorem in Dudziński [13], we can get

(3.39)

So by Lemma of Csáki and Gonchigdanzan [7] and Lemma 3.3,

(3.40)

which completes the proof.

4. Numerical Analysis

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

(41)

for finite; that is, calculate

(42)

where and

Firstly, we will construct a standardized triangular Gaussian array with equal correlation in th array for . Meanwhile, the sequence 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 to a convex sequence, is a standardized i.i.d Gaussian sequence, and is also a standardized normal random variable which is independent of . For each , let

(43)

where . Obviously, is a zero mean normal sequence with equal correlation. By this way, we get the Gaussian array needed.

Figures 1 to 3 give the actual error, , for and . In each figure, the actual error shocks tend to zero as increases. The overall performance of the actual error becomes better as .

Figure 1

T he actual error, , for and

Figure 2

T he actual error, , for and

Figure 3

T he actual error, , for and

Figures 4 to 6 give the actual error, , for

(44)

In each figure, the actual error shocks also tend to zero as increases. Also the overall performance of the actual error becomes better as .

Figure 4

T he actual error, , for and

Figure 5

T he actual error, , for and

Figure 6

T he actual error, , for and