How big is the minimum of a branching random walk?

Abstract

Let M n be the minimal position in the n-th generation, of a real-valued branching random walk in the boundary case. As n → ∞, M n − 3 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, lim sup n→∞ 1 log log log n (M n − 3 2 log n) = 1 almost surely. We also study the problem of moderate deviations of M n : P(M n − 3 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.Résumé. Soit M n la position minimaleà la n ieme génération, d'une marche aléatoire branchante réelle dans le cas frontière. Quand n → ∞, M n − 3 2 log n est tendue (voir [1, 9, 2]). Nousétablissons une loi du logarithme itéré pour décrire les limites supérieures de M n : sur l'événement de la survie du système, lim sup n→∞ 1 log log log n (M n − 3 2 log n) = 1 presque sûrement. Nousétudiionségalement les déviations modérées de M n : P(M n − 3 2 log n > λ) pour λ → ∞ et λ = o(log n). Ce problème est directement lié aux petites déviations d'une classe des cascades de Mandelbrot.Plainly Θ = |u|=1 δ {V (u)} . Let ν = Θ(R). Throughout this paper and unless stated otherwise, we shall assume that the BRW is in the boundary case, i.e.Notice that under (1.1), it is possible that P(ν = ∞) > 0. See Jaffuel [23] for detailed discussions on how to reduce a general branching random walk to the boundary case. Denote by M n := min |u|=n V (u) the minimum of the branching random walk in the nth generation (with convention: inf ∅ ≡ ∞). Hammersly [19], Kingman [24] and Biggins [7] established the law of large numbers for M n (for any general branching random walk), whereas the second order limits have attracted many recent attentions, see [1,22,9,2] and the references therein. In particular, Aïdékon [2] proved the convergence in law of M n − 3 2 log n under (1.1) and some mild conditions.On the almost sure limits of M n , it was shown in [22] that there is the following phenomena of fluctuation at the logarithmic scale. Assume (1.1). If there exists some δ > 0 such that E[ν 1+δ ] < ∞ and E R (e δx + e −(1+δ)x )Θ(dx) < ∞, then

Exact convergence rates in central limit theorems for a branching random walk with a random environment in time

“…For recent developments of the subject, see e.g. Hu and Shi [22], Shi [36], Hu [21], Attia and Barral [4] and the references therein.…”

Small deviations in lognormal Mandelbrot cascades

“…In a more general study of multifractal analysis of Mandelbrot cascade measures, Molchan [16] proved that if EW −q < ∞ for some q > 0, then also EY −2q < ∞. These results have been later improved by Liu [10] and most recently Hu [9], who have shown much stronger results relating the asymptotics of P(W ≤ x) near 0 to the asymptotics of Ee −tY near ∞ in the more general case of a smoothing transform in which the number of summands W i Y i appearing on the right hand side of (1) is random and the i.i.d. assumption on the (W i ) is relaxed.…”

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

“…They established precise estimations. On the other hand, for branching random walk, Hu in [25] studied the moderate deviation for M n − x * n + 3 2θ * log n; i.e. ; P(M n ≤ x * n − 3 2θ * log n − n ) with n = o(log n).…”

“…Motivated by [25], [23] and [17], the goal of this article is to study moderate deviation P(M n ≤ x * n − 3 2θ * log n − n ) with n = O(n). As we already mentioned, [25] first considered this problem with n = o(log n); see Remarks 1.2 and 1.5 below for more details. In particular, in Böttcher case, it was assumed in [25] that the step size is bounded.…”