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

Abstract

Let {Xn}n≥0 be a sequence of real valued random variables such that Xn=ρnXn−1+ϵn, n=1,2,…, where {(ρn,ϵn)}n≥1 are i.i.d. and independent of initial value (possibly random) X0. In this paper it is shown that, under some natural conditions on the distribution of (ρ1,ϵ1), the sequence {Xn}n≥0 is regenerative in the sense that it could be broken up into i.i.d. components. Further, when ρ1 and ϵ1 are independent, we construct a non-parametric strongly consistent estimator of the characteristic functions of ρ1 and ϵ1.