Large-time asymptotics for the density of a branching Wiener process

Révész, Pál; Rosen, Jay; Shi, Zhan

doi:10.1017/s0021900200001121

Cited by 4 publications

(7 citation statements)

References 6 publications

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“…Kabluchko (2012,[40]) gave an alternative proof of Chen's results under slightly stronger hypothesis. Révész, Rosen and Shi (2005, [35]) obtained a large time asymptotic expansion in the local limit theorem for branching Wiener processes, generalizing Chen's result.…”

Section: Introductionmentioning

confidence: 54%

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

Gao

Liu

2016

Stochastic Processes and their Applications

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International audienceChen [Ann. Appl. Probab. 11 (2001), 1242–1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process. We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form $EX(\ln^+ X)^{ 1+λ} < ∞$. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work. The results are established in the more general framework, i.e. for a branching random walk with a random environment in time. The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment

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Section: Introductionmentioning

confidence: 54%

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

Gao

Liu

2016

Stochastic Processes and their Applications

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“…It should be mentioned that in [36], the full expansion for the local limit theorem was obtained for the same model. However, Corollary 2.5 cannot be derived from the expansion in [36] (and vice versa).…”

Section: The Main Resultsmentioning

confidence: 99%

“…Gao and Liu (2016, [18]) improved and extended Chen's results on the branching Wiener process to the strongly non-lattice case under much weaker moment conditions. Révész, Rosen and Shi (2005, [36]) found full asymptotic expansions in the local limit theorem for branching Wiener processes, while Grübel and Kabluchko (2015, [22]) obtained the similar result for a branching random walk on Z and discussed the related applications in random trees. The exact convergence rate obtained in [11,18] can be formulated as the first order asymptotic expansion in the central limit theorem for the models considered therein.…”

Section: Introductionmentioning

confidence: 93%

“…The goal is twofold. On the one hand, although central limit theorems for branching random walks have been well studied and the asymptotic expansions for branching Wiener processes and lattice branching random walks were given in [36] and [22], the asymptotic expansions in central limit theorems for non-lattice branching random walks are still not known. On the other hand, we shall perform our research in a more general framework, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

Gao¹,

Liu²

2018

Bernoulli

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Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time. For the normalised counting measure of the number of particles of generation n in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central limit theorems for sums of independent random variables, truncating arguments and martingale approximation play key roles. In particular, we introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which are of independent interest.

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“…Remark 2.4. For the branching Brownian motion, an expansion similar to (19) (with β = 0) was obtained by Révész et al [35]. In a general BRW, the transition mechanism is not Gaussian, so that methods specific to the Gaussian setting cannot be used.…”

Section: 1mentioning

confidence: 99%

Edgeworth expansions for profiles of lattice branching random walks

Grübel¹,

Kabluchko²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. Consider a branching random walk on Z in discrete time. Denote by Ln(k) the number of particles at site k ∈ Z at time n ∈ N 0 . By the profile of the branching random walk (at time n) we mean the function k → Ln(k). We establish the following asymptotic expansion of Ln(k), as n → ∞:where r ∈ N 0 is arbitrary, ϕ(β) = log k∈Z e βk EL 1 (k) is the cumulant generating function of the intensity of the branching random walk andThe expansion is valid uniformly in k ∈ Z with probability 1 and the F j 's are polynomials whose random coefficients can be expressed through the derivatives of ϕ and the derivatives of the limit of the Biggins martingale at 0. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for r = 0, 1, 2 we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers Ln(kn), where kn ∈ Z depends on n in some regular way. We also prove a.s. limit theorems for the mode arg max k∈Z Ln(k) and the height max k∈Z Ln(k) of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter ϕ (0) is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.

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Large-time asymptotics for the density of a branching Wiener process

Abstract: Given an R d -valued supercritical branching Wiener process, let ψ(A, T ) be the number of particles in A ⊂ R d at time T (T = 0, 1, 2, . . . ). We provide a complete asymptotic expansion of ψ(A, T ) as T → ∞, generalizing the work of X. Chen.

Cited by 4 publications

References 6 publications

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

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

Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

Edgeworth expansions for profiles of lattice branching random walks

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