On the reflected random walk on R+

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{eqnarray*} \left\{\begin{array}{rl} X_0&=x \\ X_{n+1} & = |X_n+Y_{n+1}| \end{array}\right. \end{eqnarray*} where (Yn) 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.