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

TL;DR

This paper studies a stochastic AR(1) sequence with random coefficients, demonstrating its regenerative properties and providing a non-parametric estimator for the distributions of the coefficients.

Contribution

It establishes conditions under which the sequence is regenerative and introduces a consistent estimator for the characteristic functions of the random coefficients.

Findings

Sequence is regenerative under certain conditions.

Constructed a non-parametric estimator for coefficient distributions.

Sequence can be broken into i.i.d. components.

Abstract

Let $\{X_n\}_{n\ge0}$ be a sequence of real valued random variables such that $X_n=\rho_n X_{n-1}+\epsilon_n,~n=1,2,\ldots$, where $\{(\rho_n,\epsilon_n)\}_{n\ge1}$ 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 $\{X_n\}_{n\ge0}$ 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 $\epsilon_1$.