The many-to-few lemma and multiple spines

Harris, Simon C.; Roberts, Matthew I.

doi:10.1214/15-aihp714

Cited by 77 publications

(98 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We begin by recalling the useful technique of spinal decomposition (c.f., [7]). The (one) spine branching Brownian motion (SBBM) is defined as the original process (h, T, d), only that at any given time one of the particles is designated as the spine particle.…”

Section: Reduction To the Cluster Of The Spine Conditioned To Be Thementioning

confidence: 99%

The structure of extreme level sets in branching Brownian motion

Cortines¹,

Hartung²,

Louidor³

2019

Ann. Probab.

View full text Add to dashboard Cite

This work is a continuation of [5], in which the same authors studied the fine structure of the extreme level sets of branching Brownian motion, namely the sets of particles whose height is within a finite distance from the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. Our main finding here is that most of the particles in an extreme level set come from only a small fraction of the clusters, which are atypically large.

show abstract

Section: Reduction To the Cluster Of The Spine Conditioned To Be Thementioning

confidence: 99%

The structure of extreme level sets in branching Brownian motion

Cortines¹,

Hartung²,

Louidor³

2019

Ann. Probab.

View full text Add to dashboard Cite

show abstract

“…The proof of the following result with a detailed discussion can be found for example in [5] or [7].…”

Section: Many-to-one Lemma and Applicationsmentioning

confidence: 99%

Branching Brownian motion with spatially homogeneous and point-catalytic branching

Bocharov

Wang

2019

J. Appl. Probab.

View full text Add to dashboard Cite

show abstract

“…We can now state the main results concerning the limit of (2) with the penalization function (8). We must separate two cases for super-critical offspring distributions depending on q 0 > 0 (which is equivalent to κ > 0) or q 0 = 0 (which is equivalent to κ = 0).…”

Section: 1mentioning

confidence: 99%

Penalization of Galton–Watson processes

Abraham

Debs

2020

Stochastic Processes and their Applications

View full text Add to dashboard Cite

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton-Watson processes with a penalizing function of the form P (x)s x where P is a polynomial of degree p and s ∈ [0, 1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s → 1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton-Watson tree with p distinguished infinite spines. and the associated process Z is distributed, if 0 < κ < 1, as the size-biased process of the GW conditioned on extinction.

show abstract

The many-to-few lemma and multiple spines

Abstract: We develop a simple and intuitive identity for calculating expectations of weighted k-fold sums over particles in branching processes, generalising the well-known many-to-one lemma.

Cited by 77 publications

References 15 publications

The structure of extreme level sets in branching Brownian motion

The structure of extreme level sets in branching Brownian motion

Branching Brownian motion with spatially homogeneous and point-catalytic branching

Penalization of Galton–Watson processes

Contact Info

Product

Resources

About