Reversal property of the Brownian tree

Abraham, Romain; Delmas, Jean‐François

doi:10.30757/alea.v15-48

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Local Times and Other Contour Processes1

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2021

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“…where ζ i is the height of the tree τ i . This result can also be deduced from the reversal property of the Brownian tree; see [3].…”

Section: Local Times and Other Contour Processesmentioning

confidence: 74%

Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length

Abraham

Delmas

2021

Adv. Appl. Probab.

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We consider a model of a stationary population with random size given by a continuous-state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure for the genealogical tree of n individuals randomly chosen among the extant population at a given time. Then we prove the convergence of the renormalized total length of this genealogical tree as n goes to infinity; see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant-size population. The limit appears already in Bi and Delmas (2016) but with a different approximation of the full genealogical tree. The proof is based on the ancestral process of the extant population at a fixed time, which was defined by Aldous and Popovic (2005) in the critical case.

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“…where ζ i is the height of the tree τ i . This result can also be deduced from the reversal property of the Brownian tree; see [3].…”

Section: Local Times and Other Contour Processesmentioning

confidence: 74%