Limit theorems for auto regression processwith random parameter v, 0 < v < 1

Abstract

In this paper we obtain the criterion of weak convergence of the sequence of the sum of the first $n$ terms of the linear process $\left\{X_{kn} ,\; k=1,2,...,n;\; n=1,2,...\right\}$ with random coefficients $\left\{v^{k} ,k\in {\mathbb N}\right\}$, generated by the innovation sequence $\left\{\xi _{kn} ,k\in Z\right\}$ satisfying the condition of infinite smallness to the limit distribution and as a consequence of this result we obtain the analog of the Lindeberg-Feller theorem for the auto regression process with random parameter $v,\; 0{\rm \; }<{\rm \; }v<1$. In addition, the strong law of large numbers and the law of iterated logarithm are proved.