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Reward distributions associated with some block tridiagonal transition matrices with applications to identity by descent

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Valeri T. Stefanov  and
Frank Ball
Valeri T. Stefanov*
Affiliation:
The University of Western Australia
Frank Ball*
Affiliation:
University of Nottingham
*
∗ Postal address: School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia. Email address: stefanov@maths.uwa.edu.au
∗∗ Postal address: School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK. Email address: fgb@maths.nott.ac.uk
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Abstract

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Markov and semi-Markov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships.

Keywords

Semi-Markov processstopping timereward functionLaplace transformgenomic continuum model

MSC classification

Primary: 60E10: Characteristic functions; other transforms
Secondary: 92D10: Genetics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 523 - 545
Copyright
Copyright © Applied Probability Trust 2009 

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