Local behaviour of first passage probabilities

2010/06/28 by R. A. Doney, Ronald Doney, Doney, Ronald · 2 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1006.5316

arxiv created 2010/06/28 · openalex publication_date 2010/06/28 · arxiv updated 2010/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that S is an asymptotically stable random walk with norming sequence c_n and that T_x is the time that S first enters (x,inf), where x≥ 0. The asymptotic behaviour of P(T₀=n) has been described in a recent paper of Vatutin and Wachtel, \citevw, and here we build on that result to give three estimates for P(T_x=n), which hold uniformly as n→inf in the regions x=o(c_n), x=O(c_n), and x/c_n→inf, respectively.

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• A local limit theorem for lattice oscillating random walks
• Reduced critical branching processes in non-favorable random environment

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