## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 3 (1990), 1159-1173.

### Limiting Distributions of Nonlinear Vector Functions of Stationary Gaussian Processes

Hwai-Chung Ho and Tze-Chien Sun

#### Abstract

Given a stationary Gaussian vector process $(X_m, Y_m), m \in Z$, and two real functions $H(x)$ and $K(x)$, we define $Z^n_H = A^{-1}_n\sum^{n - 1}_{m = 0} H(X_m)$ and $Z^n_K = B^{-1}_n\sum^{n - 1}_{m = 0} K(Y_m)$, where $A_n$ and $B_n$ are some appropriate constants. The joint limiting distribution of $(Z^n_H, Z^n_K)$ is investigated. It is shown that $Z^n_H$ and $Z^n_K$ are asymptotically independent in various cases. The application of this to the limiting distribution for a certain class of nonlinear infinite-coordinated functions of a Gaussian process is also discussed.

#### Article information

**Source**

Ann. Probab., Volume 18, Number 3 (1990), 1159-1173.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990740

**Digital Object Identifier**

doi:10.1214/aop/1176990740

**Mathematical Reviews number (MathSciNet)**

MR1062063

**Zentralblatt MATH identifier**

0712.60021

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G15: Gaussian processes

**Keywords**

Central limit theorem noncentral limit theorem long-range dependence stationary Gaussian vector processes

#### Citation

Ho, Hwai-Chung; Sun, Tze-Chien. Limiting Distributions of Nonlinear Vector Functions of Stationary Gaussian Processes. Ann. Probab. 18 (1990), no. 3, 1159--1173. doi:10.1214/aop/1176990740. https://projecteuclid.org/euclid.aop/1176990740