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July 2018 Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption
Ion Grama, Ronan Lauvergnat, Émile Le Page
Ann. Probab. 46(4): 1807-1877 (July 2018). DOI: 10.1214/17-AOP1197

Abstract

Consider a Markov chain $(X_{n})_{n\geq0}$ with values in the state space $\mathbb{X}$. Let $f$ be a real function on $\mathbb{X}$ and set $S_{n}=\sum_{i=1}^{n}f(X_{i})$, $n\geq1$. Let $\mathbb{P}_{x}$ be the probability measure generated by the Markov chain starting at $X_{0}=x$. For a starting point $y\in\mathbb{R}$, denote by $\tau_{y}$ the first moment when the Markov walk $(y+S_{n})_{n\geq1}$ becomes nonpositive. Under the condition that $S_{n}$ has zero drift, we find the asymptotics of the probability $\mathbb{P}_{x}(\tau_{y}>n)$ and of the conditional law $\mathbb{P}_{x}(y+S_{n}\leq \cdot\sqrt{n}\mid\tau_{y}>n)$ as $n\to+\infty$.

Citation

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Ion Grama. Ronan Lauvergnat. Émile Le Page. "Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption." Ann. Probab. 46 (4) 1807 - 1877, July 2018. https://doi.org/10.1214/17-AOP1197

Information

Received: 1 July 2016; Revised: 1 February 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919013
MathSciNet: MR3813980
Digital Object Identifier: 10.1214/17-AOP1197

Subjects:
Primary: 60F05 , 60G50 , 60J50
Secondary: 60G40 , 60J05 , 60J70

Keywords: Exit time , Harmonic function , limit theorem , Markov chain , Random walk

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 4 • July 2018
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