Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications

Zhicheng Chen,Xinsheng Liu

doi:10.1155/2020/3945946

Zhicheng Chen, Xinsheng Liu

Open Access

https://doi.org/10.1155/2020/3945946

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Abstract

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.

Highlights

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. e simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples
Lacey and Philipp prove that equation (1) holds when f is a bound Lipschitz function [8]. e almost sure central limit theorem on maximum of i.i.d random variables is firstly discovered by Fahrnar and Stadtmuller and Cheng et al, respectively [5, 9]
Peng et al extend the result to the maxima and minima of the complete and incomplete samples for weakly dependent stationary Gaussian sequences [16]. e almost sure central limit theorem (ASCLT) for the maxima and sums of i.i.d. random variables is investigated by Zang et al [17]

Summary

Research Article

Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications. The almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. Chen and Peng provide an ASCLT for the maxima of multivariate stationary Gaussian sequences under some mild conditions [13, 14]. Tan and Wang consider ASCLT for the maxima and sums of standardized stationary Gaussian sequences under some conditions [18]. E behavior of almost sure convergence of the maxima and the minima for a strongly dependent stationary Gaussian sequence is extended to multivariate vectors [20, 21]. We study ASCLTof the multivariate general standard normal sequences under some suitable conditions. Roughout this paper, 􏼈X1, X2, . . .􏼉 is a standardized normal sequence of d-dimensional random vector, i.e., each

Results

We shall be concerned with the maxima

Theorem mensional