On the reflected random walk on $R_{+}$

Boyer, Jean−Baptiste

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 350-368.

Let $\rho$ be a borelian probability measure on $R$ having a moment of order $1$ and a drift $\lambda =\int {}_{R}yd\rho \left(y\right)<0$. Consider the random walk on $R_{+}$ starting at $x\in R_{+}$ and defined for any $n\in N$ by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill X_0& =x\hfill \\ \hfill X_{n+1}& =|X_n+Y_{n+1}|\hfill \end{array}\right.\end{array}$$

where $\left(Y_n\right)$ is an iid sequence of law $\rho$. 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 $\rho$ 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.

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