On Convergence of Weakly Dependent Processes to the Wiener Process

A. A. Borovkov

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 193-221

Résumé

Let $R(0,T)$ be a complete separable metric space of measurable real-valued functions on $[0,T]$. We give same conditions distinguishing a wide class of processes $\{Z_T(t),\ 0\le t\le T\}$ on $R(0,T)$ which are in some way close to Markov processes (for large $T$) and are such that the distributions of functionals $f$, continuous in the uniform metric, on $Z_T(t)$ converge as $T\to\infty$ to the distributions of the functionals $f(w)$ on the Wiener process $w(t)$. Roughly speaking, the essence of this conditions is as follows. There must exist a recurrent set $D$ of “states” of the process (in some cases of a certain other process, a function of which is the one under consideration) such that the mean and variance of the increment of the process for a long period of time $t$ after hitting $D$ are asymptotically as $t\to\infty$ independent of the history of the path before hitting $D$. Moreover it is required that the time taken to return to $D$ have a uniformly bounded moment of order $1+\gamma$, $\gamma>0$, and the “swings” of the trajectories between returns to $D$ have a uniformly bounded moment of order $2+\gamma$ . These conditions seem to be the most convenient, for example, far a number of problems connected with semi-Markovian processes and various generalized renewal processes arising in queueing theory. Examples are also given in the paper.