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

Valeri Stefanov, F. Ball

    Research output: Contribution to journalArticlepeer-review

    Abstract

    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.
    Original languageEnglish
    Pages (from-to)523-545
    JournalAdvances in Applied Probability
    Volume41
    Issue number2
    DOIs
    Publication statusPublished - 2009

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