How big is the minimum of a branching random walk?

2013/05/28 by Hu, Yueyun · 2 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1305.6448

Abstract

Let M_n be the minimal position at generation n, of a real-valued branching random walk in the boundary case. As n → ∞, M_n- 3 \over 2 log n is tight (see [1][9][2]). We establish here a law of iterated logarithm for the upper limits of M_n: upon the system's non-extinction, \limsup_n→ ∞ 1\over log log log n ( M_n - 3\over2 log n) = 1 almost surely. We also study the problem of moderate deviations of M_n: p(M_n- 3 \over 2 log n > λ) for λ→ ∞ and λ=o(log n). This problem is closely related to the small deviations of a class of Mandelbrot's cascades.

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