On the lattice case of an almost-sure renewal theorem for branching random walks

Gatzouras, D.

doi:10.1017/s0001867800010223

Cited by 9 publications

(15 citation statements)

References 7 publications

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“…This martingale, which is called the intrinsic martingale in the BRW, is of outstanding importance in the asymptotic analysis of the BRW (see e.g. [8] and [14]). In this article, we give sufficient conditions for the following statement to hold: for fixed a > 0 n≥0 e an (W − W n ) converges a.s.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

Iksanov

Meiners

2010

Journal of Applied Probability

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We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. The case of Galton-Watson processes is particularly included so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering. As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be interesting on its own and which correct, generalize and simplify some earlier works.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

Iksanov

Meiners

2010

Journal of Applied Probability

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“…Coupled with Lemma 3.1, this result hints that we should not expect to have a central limit theorem when z φ (t) − z φ (∞) does not decay at least as fast as e −γt/2 , the threshold for the second term in the right-hand side of (20) to converge to 0. We will explain more precisely why this is sharp in Remark 4.4.…”

Section: 3mentioning

confidence: 78%

“…This lemma and the decomposition in (20) show that understanding the rate of convergence of z φ (t) to its limit in the renewal theorem is helpful for estimating the fluctuations of Z φ .…”

Section: The Central Limit Theorem For ∆ N -General Branching Processesmentioning

confidence: 95%

Central limit theorems for the spectra of classes of random fractals

Charmoy¹,

Croydon

Hambly³

2017

Trans. Amer. Math. Soc.

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Abstract. We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem. The Brownian continuum random tree can also be viewed as a random fractal generated by a Dirichlet distribution. The first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this, though the positivity of the variance arising in the central limit theorem is left open. In both cases these fractals can be described through a general Crump-Mode-Jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics. Our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known. MSC: 28A80 (primary), 60J80, 35P20 (secondary).

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“…Comparing Theorem 1.1 and Theorem 1.4, we remark that the almost sure behaviors of M n are not related to the moderate deviations of M n . This can be explained as follows: Define for all λ ≥ 0 and u ∈ T, 18) where here and in the sequel, {u 0 = ∅, u 1 , ..., u |u| := u} denotes the shortest path from ∅ to u such that |u i | = i for all 0 ≤ i ≤ |u|. We introduce the stopping lines:…”

Section: Introductionmentioning

confidence: 99%

“…We are very grateful to two anonymous referees for their careful readings and helpful comments on the first version of this paper. We also thank Zhan Shi for the reference [18].…”

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confidence: 99%

How big is the minimum of a branching random walk?

Hu¹

2016

Ann. Inst. H. Poincaré Probab. Statist.

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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

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On the lattice case of an almost-sure renewal theorem for branching random walks

Cited by 9 publications

References 7 publications

Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

Central limit theorems for the spectra of classes of random fractals

How big is the minimum of a branching random walk?

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