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.