The invariance principle for stationary processes

Yu. A. Davydov

Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 3, pp. 498-509

Résumé

Let $X_t$ be a stationary process, $\mathbf EX_t=0$, $\mathbf DX_t=\sigma^2\infty$, $S_T=\sum_1^TX_t$ (discrete time), or $S_t=\int_0^TX_t\,dt$ (continuous time). Define $X_T(t)$ as in (1) or (2). Let $B_T(s,t)$ be the covariance function of $X_t(t)$. Let distribution $P_T$ correspond to the process $X_T(t)$ and distribution $W_\gamma$ correspond to a Gaussian process with the covariance function $$ B_\gamma(s,t)=\frac12(s+t+|s^{1/\gamma}-t^{1/\gamma}|^\gamma). $$ Theorem 1. If $\psi(T)=\mathbf DS_T\uparrow\infty$, $P_T\Rightarrow W_\gamma$, then $B_T(s,t)\to B_\gamma(s,t)$ and $\psi(T)=T^\gamma h(T)$, where $h(T)$ is a slowly changing function. Theorem 2. {\em Let $X_j=\sum_{i=-\infty}^\infty c_{i-j}\xi_i$ where $\xi_i$ are independent identically distributed random variables, $\mathbf E\xi_i=0$, $\mathbf E\xi_i^{2k}\infty$ and $\sum c_j^2\infty$. If $\mathbf DS_n=n^\gamma h(n)$, $2/(k+2)\gamma\le2$, where $h(n)$ is a slowly changing function, then $P_n\Rightarrow W_\gamma$.} In the next two theorems the invariance principle is proved for processes generated by mixing processes.