## The Annals of Probability

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

#### 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

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