Valeri T. Stefanov

A note on integrals for birth-death processes

A general integral for birth-death Markov processes is considered. A closed form expression is provided for its expected value in terms of the birth and death rates. A simple route for numerical evaluation of its variance is also suggested.

Explicit asymptotic results for open times in ion channel models

The dynamical aspects of single channel gating can be modelled by a Markov renewal process, with states aggregated into two classes corresponding to the receptor channel being open or closed, and with brief sojourns in either class not detected. This paper is concerned with the relation between the amount of time, for a given record, in which the channel appears to be open compared to the amount in which it is actually open and the difference in their proportions; this may be used to obtain information on the unobserved actual process from the observed one. Results, with extensions, on exponential families have been applied to obtain relevant generating functions and asymptotic normal distributions, including explicit forms for the parameters. Numerical results are given as illustration in special cases.

REWARD DISTRIBUTIONS ASSOCIATED WITH SOME BLOCK TRIDIAGONAL TRANSITION MATRICES WITH APPLICATIONS TO IDENTITY BY DESCENT

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.

ON THE MOMENTS OF SOME FIRST-PASSAGE TIMES

Let {X,}),o (t may be discrete or continuous) be a random process whose finite-dimensional distributions are of exponential type. The first-passage time inf{t:X, f(t)}, where f(t) is a positive, continuous function, such that f(t)= o(t) as t t 00, is considered. The problem of finiteness of its moments is solved for both the case that {X,}to has stationary independent increments as well as the case in which no assumptions are made about stationarity and independence for the increments of the process. Applications to sequential estimation are also given.