Lower deviation probabilities for level sets of the branching random walk

2020/12/02 by Shuxiong Zhang, Zhang, Shuxiong
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2012.00911

openalex publication_date 2020/12/02 · openalex created_date 2022/11/25 · openalex updated_date 2026/07/28

Abstract

Given a branching random walk\Z_n\_n≥0 on ℝ, let Z_n([y,∞)) be the number of particles located in [y,∞) at generation n. It is known from \citeBiggins1977 that under some mild conditions, n^-1log Z_n([θx^* n,∞)) converges a.s. to log m-I(θx^*), where log m-I(θx^*) is a positive constant. In this work, we investigate its lower deviation, in other words, the convergence rates of ℙ(Z_n([θx^* n,∞))

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