On the reflected random walk on $\mathbb{R}_+$.
Résumé
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.
Origine : Fichiers produits par l'(les) auteur(s)
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