Dynamical Borel-Cantelli lemma for autoregressive processes with Laplace noises

Abstract

Consider a deterministic dynamical system (M, F, µ, T), where µ is T−invariant probability measure. The well-known dynamical Borel-Cantelli lemma states that for certain sequences of measurable subsets A n ⊂ M and µ− almost every point x the inclusion T n x ∈ A n holds for infinitely many values n. In the present paper, we study the stationary Markov process X := {X n , n ∈ N} defined as X n := X n (ρ,ξ) = ρX n−1 + ξ n , n ∈ Z, where ρ is a real constant, ξ := {ξ n ,n ∈ Z} is a sequence of independent, identically distributed (i.i.d.) random variables and ξ 0 ∼ Laplace(0,b). Let (R Z , B, ν) be the probability space, where ν is a probability measure associated by stochastic process X. Consider the shift map τ on R Z . We give sufficient conditions on sequences of cylinders, that ensure the dynamical Borel-Cantelli lemma for the dynamical system (R Z , B,ν,τ). It also holds for AR(1) processes generated by the exponential, uniform, and Laplace distributions