Upper deviation probabilities for level sets of a supercritical branching random walk

2024/02/06 by Zhang, Shuxiong, Luo, Lianghui
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2402.03872

Abstract

Given a supercritical branching random walk \Z_n\_{n≥ 0} on ℝ, let Z_n([y,∞)) be the number of particles located in [y,∞)⊂ℝ at generation n. Let m be the mean of the offspring law of \Z_n\_{n≥ 0} and I(x) be the large deviation rate function of the underlying random walk of \Z_n\_{n≥ 0}. It is known from [6] that under some mild conditions, for x∈(0,x^*), n^-1log Z_n([nx,∞)) converges almost surely to log m- I(x) on the event of nonextinction as n→∞, where x^* is the speed of maximal position of the branching random walk. In this work, we investigate its upper deviation probabilities, in other words, the convergence rates of ℙ(Z_n([xn,∞))≥ e^an) as n→∞, where x>0 and a>(log m- I(x))⁺. This paper is a counterpart work of the lower deviation probabilities [28] and also completes those results in [1] for the branching Brownian motion.

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