On the reflected random walk on $\mathbb{R}_+$.

Let $\rho$ be a borelian probability measure on $\mathbb{R}$ having a moment of order $1$ and a drift $\lambda = \int_{\mathbb{R}} y\mathrm{d}\rho(y)<0$. Consider the random walk on $\mathbb{R}_+$ starting at $x\in \mathbb{R}_+$ and defined for any $n\in \mathbb{N}$ by \[ \left\{\begin{array}{rl} X_0&=x \\ X_{n+1} & = |X_n+Y_{n+1}| \end{array}\right. \] where $(Y_n)$ is an iid sequence of law $\rho$. We note $P$ the Markov operator associated to this random walk. This is the operator defined for any borelian and bounded function $f$ on $\mathbb{R}_+$ and any $x\in \mathbb{R}_+$ by \[ Pf(x) = \int_{\mathbb{R}} f(|x+y|) \mathrm{d} \rho(y) \] For a borelian bounded function $f$ on $\mathbb{R}_+$, 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 $\rho$, for any directly Riemann-integrable function, there is a solution to Poisson's equation and using the renewal theorem, we prove that this solution has a limit at infinity. Then, we use this result to prove the law of large numbers, the large deviation principle, the central limit theorem and the law of the iterated logarithm.