On the reflected random walk on $\mathbb{R}_+$.
| hal.structure.identifier | Université Paris Nanterre [UPN] | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | BOYER, Jean-Baptiste | |
| dc.date.accessioned | 2024-04-04T03:14:55Z | |
| dc.date.available | 2024-04-04T03:14:55Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/194107 | |
| dc.description.abstractEn | 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. | |
| dc.language.iso | en | |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/ | |
| dc.subject.en | Markov chains | |
| dc.subject.en | Poisson's equation | |
| dc.subject.en | Gordin's method | |
| dc.subject.en | Renewal theorem | |
| dc.subject.en | Random walk on the half-line | |
| dc.title.en | On the reflected random walk on $\mathbb{R}_+$. | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Probabilités [math.PR] | |
| dc.identifier.arxiv | 1506.07623 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-01167918 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01167918v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=BOYER,%20Jean-Baptiste&rft.genre=preprint |
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