AR(1) sequence with random coefficients: Regenerative properties and its application

Abstract

Let $\{X_n{\}}_{n\ge 0}$ be a sequence of real valued random variables such that $X_n={\rho }_n{X}_{n-1}+{\epsilon }_n,n=1,2,\dots$, where $\{({\rho }_n,{\epsilon }_n){\}}_{n\ge 1}$ are i.i.d. and independent of initial value (possibly random) $X_0$. In this paper it is shown that, under some natural conditions on the distribution of $({\rho }_1,{\epsilon }_1)$, the sequence $\{X_n{\}}_{n\ge 0}$ is regenerative in the sense that it could be broken up into i.i.d. components. Further, when ${\rho }_1$ and ${\epsilon }_1$ are independent, we construct a non-parametric strongly consistent estimator of the characteristic functions of ${\rho }_1$ and ${\epsilon }_1$.