Convergence results on multitype, multivariate branching random walks

Biggins, J. D.; Sani, Alireza

doi:10.1017/s0001867800000422

Cited by 15 publications

(29 citation statements)

References 19 publications

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“…Similar results were obtained by many authors in special cases and in different contexts; see, for example, [16], [27], [32] for multiplicative cascades, [14], [15] for branching process and the Crump-ModeJagers process, [2], [8], [9], [12] for branching random walks, [19] for smoothing processes, [36] for random fractals, and [26] for branching processes in random environments.…”

Section: Resultssupporting

confidence: 85%

“…Under this assumption, all particles u ∈ I associated with the number of their offspring N u form a Galton-Watson tree T. We remark that this assumption is not necessary in usual MBRWs, so the example presented here is just a special case of an MBRW. For more information and results about the usual MBRW; see [10], [12], and [29].…”

Section: Huangmentioning

confidence: 99%

“…The model considered here is a generalization of the model presented in [32] to the multidimensional case. Similar to the one-dimensional case, our model corresponds to a multitype branching random walk, 2 C. HUANG which has attracted recent attention in the literature; see, for example, Biggins [10], Biggins and Rahimzadeh Sani [12], and Kyprianou and Rahimzadeh Sani [29].…”

Section: Introductionmentioning

confidence: 99%

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Moments for multidimensional Mandelbrot cascades

Huang¹

2016

J. Appl. Probab.

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We consider the distributional equationwhere N is a random variable taking value in N 0 = {0, 1, . . .}, A 1 , A 2 , . . . are p × p nonnegative random matrices, and Z, Z(1), Z(2), . . . , are independent and identically distributed random vectors in R p + with R + = [0, ∞), which are independent of (N, A 1 , A 2 , . . .). Let {Y n } be the multidimensional Mandelbrot martingale defined as sums of products of random matrices indexed by nodes of a Galton-Watson tree plus an appropriate vector. Its limit Y is a solution of the equation above. For α > 1, we show a sufficient condition for E Y α ∈ (0, ∞). Then for a nondegenerate solution Z of the distributional equation above, we show the decay rates of Ee −t·Z as t → ∞ and those of the tail probability P(y ·Z ≤ x) as x → 0 for given y = (y 1 , . . . , y p ) ∈ R p + , and the existence of the harmonic moments of y ·Z. As an application, these results concerning the moments (of positive and negative orders) of Y are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrices of the equation above are complex, a sufficient condition for the L α convergence and the αth-moment of the Mandelbrot martingale {Y n } are also established.

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Section: Resultssupporting

confidence: 85%

Section: Huangmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Moments for multidimensional Mandelbrot cascades

Huang¹

2016

J. Appl. Probab.

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“…With motivations from [12], [33], and the works mentioned above, we analyze the point process induced by a branching random walk with heavy-tailed displacements, continuing the research initiated by Bhattacharya et al [10], and generalize it to the multitype case, allowing dependence between weights. More formally, in this paper we consider a multitype branching random walk with heavy tailed increments.…”

Section: Introductionmentioning

confidence: 99%

Extremes of multitype branching random walks: heaviest tail wins

et al. 2019

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We consider a branching random walk on a multi(Q)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index α, while the other types of particles have lighter tails than that of particles of type Q. In this article, we derive the weak limit of the sequence of point processes associated with the positions of the particles in the n th generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process (SScDPPP) using the tools developed in [10]. As a consequence, we shall obtain the asymptotic distribution of the position of the rightmost particle in the n th generation.

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“…[17], [18], [19], [20], and [22]) along with some variants of the process (see [2], [3], [4], [5], and [7]). The discrete-time case has been initially considered as a natural generalization of branching processes (see [1], [10], [11], [12], [13], and [16]). The definition of a discrete-time branching random walk (BRW) that we give in Section 2.1 is sufficiently general to include the discrete-time counterpart that every continuoustime BRW admits.…”

Section: Introductionmentioning

confidence: 99%

Strong Local Survival of Branching Random Walks is Not Monotone

Bertacchi

Zucca

2014

Advances in Applied Probability

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In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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Convergence results on multitype, multivariate branching random walks

Cited by 15 publications

References 19 publications

Moments for multidimensional Mandelbrot cascades

Moments for multidimensional Mandelbrot cascades

Extremes of multitype branching random walks: heaviest tail wins

Strong Local Survival of Branching Random Walks is Not Monotone

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