BOYER, Jean-Baptiste

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BOYER, Jean-Baptiste
Université Paris Nanterre [UPN]
Institut de Mathématiques de Bordeaux [IMB]

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Document de travail - Pré-publication

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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.< Réduire

Mots clés en anglais

Markov chains

Poisson's equation

Gordin's method

Renewal theorem

Random walk on the half-line

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

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Résumé en anglais

Mots clés en anglais

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