Volume 21, 2017
|Page(s)||350 - 368|
|Published online||12 December 2017|
On the reflected random walk on R+
IMB, Université de Bordeaux / MODAL’X, Université Paris-Ouest, Nanterre, France
Received: 29 April 2016
Revised: 24 November 2016
Accepted: 12 May 2017
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 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.
Mathematics Subject Classification: 60J10
Key words: Markov chains / Poisson’s equation / Gordin’s method / renewal theorem / random walk on the half line
© EDP Sciences, SMAI, 2017
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