Lower deviation and moderate deviation probabilities for maximum of a branching random walk

2018/07/22 by Chen, Xinxin, He, Hui · 2 citations
#60F10 #60J60 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1807.08263

Abstract

Given a super-critical branching random walk on \mathbb R started from the origin, let M_n be the maximal position of individuals at the n-th generation. Under some mild conditions, it is known from \citeA13 that as n→∞, M_n-x^*n+(3)/(2θ^*)log n converges in law for some suitable constants x^* and θ^*. In this work, we investigate its moderate deviation, in other words, the convergence rates of ℙ(M_n≤ x^*n-(3)/(2θ^*)log n-ℓ_n), for any positive sequence (ℓ_n) such that ℓ_n=O(n) and ℓ_n\uparrow∞. As a by-product, we also obtain lower deviation of M_n; i.e., the convergence rate of ℙ(M_n≤ xn), for x

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