International Conference on Distribution Theory, Order Statistics, and Inference (2004. Universidad de Cantabria.)

Advances in Distribution Theory, Order Statistics, and Inference

In this paper, we investigate sooner and later waiting time problems for patterns So and S 1 in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences of So and between successive occurrences of S 0 and S 1 and of the waiting time until the rth occurrence of So are also given.

Sooner and later waiting time problems for patterns in Markov dependent trials

The expected value and generating function of the number of overlapping occurrences of a pattern P in a Markov chain until the first occurrence of a member from a finite collection of patterns that start with P is obtained. A martingale technique is employed to address the problem.

/pdf/on-occurrence-of-subpattern-and-method-of-gambling-teams-27eoktibdj.pdf

On occurrence of subpattern and method of gambling teams

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.

/pdf/reward-distributions-associated-with-some-block-tridiagonal-5b7vcq91yl.pdf

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

Let a discrete time semi-Markov process Z ∈ with finite state space an alphabet Defining the process U ∈ to be the backward recurrence time of the process Z ∈ we study the Markov process Z U ∈ We give its transition probabilities of first and higher order, the limiting distribution, and the stationary distribution. Using this Markov process we construct a k-dimensional process Z U ∈ and we study its basic properties. As an application we consider a finite set of words W = w1 w2 w of equal length k which are produced under the semi-Markovian hypothesis and we focus on the waiting time for the first word occurrence from the set W . The corresponding probability distribution, the generating function, as well as the mean waiting time and variance are obtained.

Applied Probability On Discrete Time Semi-Markov Chains and Applications in Words Occurrences

Some waiting time problems

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Some waiting time problems