Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 185, Issue 3, Pages -Publisher
SPRINGER
DOI: 10.1007/s10955-021-02833-y
Keywords
Branching random walk; Galton-Watson tree; Gap statistics; Conditioned on particle numbers
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This paper investigates a pruned Galton-Watson tree and its branching random walk, providing an explicit probability measure for the first n generations of the tree and the asymptotic tail behavior of span and gap statistics of its k particles. This study extends the work of Ramola et al. (Chaos Solitons Fractals 74:79-88, 2015) to arbitrary offspring and displacement distributions with moment constraints.
In this paper, we consider a pruned Galton-Watson tree conditioned to have k particles in generation n, i.e. we take a Gallon-Watson tree satisfying Z(n) = k, and delete all branches that die before generation n. We show that with k fixed and n -> infinity, the first n generations of this tree can be described by an explicit probability measure P-k(st). As an application, we study a branching random walk (V-u)(u is an element of T) indexed by such a pruned Galton-Watson tree T, and give the asymptotic tail behavior of the span and gap statistics of its k particles in generation n, (V-u)(vertical bar u vertical bar=n). This is the discrete version of Ramola et al. (Chaos Solitons Fractals 74:79-88, 2015), generalized to arbitrary offspring and displacement distributions with moment constraints.
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