Conditioned local limit theorems for random walks defined on finite Markov chains

Let \((X_n)_{n\geq 0}\) be a Markov chain with values in a finite state space \(\mathbb X\) starting at \(X_0=x \in \mathbb X\) and let \(f\) be a real function defined on \(\mathbb X\). Set \(S_n=\sum_{k=1}^{n} f(X_k)\), \(n\geqslant 1\). For any \(y \in \mathbb R\) denote by \(\tau_y\) the first t...

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Bibliographic Details
Published in:arXiv.org 2017-07
Main Authors: Grama, Ion, Lauvergnat, Ronan, Emile Le Page
Format: Article
Language:English
Subjects:
Asymptotic properties
Conditioning
Markov analysis
Markov chains
Random walk
Random walk theory
Theorems
Online Access:Get full text
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Summary:Let \((X_n)_{n\geq 0}\) be a Markov chain with values in a finite state space \(\mathbb X\) starting at \(X_0=x \in \mathbb X\) and let \(f\) be a real function defined on \(\mathbb X\). Set \(S_n=\sum_{k=1}^{n} f(X_k)\), \(n\geqslant 1\). For any \(y \in \mathbb R\) denote by \(\tau_y\) the first time when \(y+S_n\) becomes non-positive. We study the asymptotic behaviour of the probability \(\mathbb P_x \left( y+S_{n} \in [z,z+a] \,,\, \tau_y > n \right)\) as \(n\to+\infty.\) We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order \(n^{3/2}\) and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability \(\mathbb P_x \left( \tau_y = n \right)\) as \(n\to+\infty\).
ISSN:2331-8422