Diffusion processes and coalescent trees
Griffiths, Robert C. and Spanò, Dario (2009) Diffusion processes and coalescent trees. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Vol.2009 (No.40).
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We dedicate this paper to Sir John Kingman on his 70th Birthday.
In modern mathematical population genetics the ancestral history of
a population of genes back in time is described by John Kingman’s
coalescent tree. Classical and modern approachesmodel gene frequencies
by diffusion processes. This paper, which is partly a review, discusses
how coalescent processes are dual to diffusion processes in an analytic
and probabilistic sense.
Bochner (1954) and Gasper (1972) were interested in characterizations
of processes with Beta stationary distributions and Jacobi polynomial
eigenfunctions. We discuss the connection with Wright-Fisher diffusions
and the characterization of these processes. Subordinated Wright-Fisher
diffusions are of this type. An Inverse Gaussian subordinator is interesting and important in subordinated Wright-Fisher diffusions and is
related to the Jacobi Poisson Kernel in orthogonal polynomial theory.
A related time-subordinated forest of non-mutant edges in the Kingman
coalescent is novel.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Diffusion processes, Population genetics -- Mathematical models|
|Series Name:||Working papers|
|Publisher:||University of Warwick. Centre for Research in Statistical Methodology|
|Place of Publication:||Coventry|
|Number of Pages:||23|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
 Andrews, G. E., Askey, R and Roy, R. (1999). Special functions, Encyclope-
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