On the reflected random walk on R₊

IMB, Université de Bordeaux / MODAL’X, Université Paris-Ouest, Nanterre, France
jeaboyer@math.cnrs.fr

Received: 29 April 2016
Revised: 24 November 2016
Accepted: 12 May 2017

Abstract

Let ρ be a borelian probability measure on R having a moment of order 1 and a drift λ = ∫_Rydρ(y) < 0. Consider the random walk on R₊ starting at x ∈ R₊ and defined for any n ∈N by $\begin{array}{ccc}\{\begin{array}{c}\\ & \\ & \end{array}& & \end{array}$ where (Y_n) is an iid sequence of law ρ. We denote P the Markov operator associated to this random walk and, for any borelian bounded function f on R₊, we call Poisson’s equation the equation f = g − Pg with unknown function g. In this paper, we prove that under a regularity condition on ρ and f, there is a solution to Poisson’s equation converging to 0 at infinity. Then, we use this result to prove the functional central limit theorem and it’s almost-sure version.