Some limit theorems for the processes with random time

A. N. Borodin

Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 754-770

Résumé

Suppose that $G(t,\omega)$ and $F(t,\omega)$ are independent stochastic processes satisfying Rosenblatt's mixing condition (0.3). We consider the processes with random time $Q(t,\omega)=G(H(t),\omega)$, where the function $H(t)$ in the case of continuous $t$ is defined by (0.8) and in the case of discrete $t$ – by (0.9). The weak convergence of the process $$ Z_{\varepsilon}(\tau)=\varepsilon^{1/2}\int_0^{\tau/\varepsilon}G(H(t),\omega)\,dt\qquad(0\le\tau\le 1) $$ to the process $\sqrt{V(\omega)}W(\tau)$ is proved. Here $V(\omega)$ is defined by (3.2) and $W(\tau)$ is a Wiener process independent of the random variable $V(\omega)$. A stochastic approximation procedure for the processes with random time is discussed also.