On the reflected random walk on R +
ESAIM: Probability and Statistics, Tome 21 (2017), pp. 350-368.

Let ρ be a borelian probability measure on R having a moment of order 1 and a drift λ= R ydρ(y)<0. Consider the random walk on R + starting at xR + and defined for any nN by

X 0 =xX n+1 =|X n +Y n+1 |
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.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2017012
Classification : 60J10
Mots clés : Markov chains, Poisson’s equation, Gordin’s method, renewal theorem, random walk on the half line
Boyer, Jean−Baptiste 1

1 IMB, Université de Bordeaux / MODAL’X, Université Paris-Ouest, Nanterre, France
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Boyer, Jean−Baptiste. On the reflected random walk on $R_{+}$. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 350-368. doi : 10.1051/ps/2017012. http://archive.numdam.org/articles/10.1051/ps/2017012/

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