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

Abstract

Let a discrete time semi-Markov process Z ∈ with finite state space an alphabetDefining 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 = w 1 w 2 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.

On the number of word occurrences in a semi-Markov sequence of letters

“…Furthermore, Chryssaphinou et al [2] defined {U γ ; γ ∈ N} the backward recurrence time for the semi-Markov process…”

“…We also present some theorems, which have been proved by Chryssaphinou et al [2] concerning the Markov process {(Z γ , U γ ); γ ∈ N} and are used in order to prove the new results of this paper. In Section 2, we derive the probability of a word occurrence in a DTSM model.…”

“…Theorem 1.6. (see [2]) Let the DTMRP {(J n , S n ); n ∈ N} with finite state space Ω be irreducible, aperiodic…”

“…Chryssaphinou et al [2] defining the process {U γ ; γ ∈ N} to be the backward recurrence times of the DTSM process {Z γ ; γ ∈ N} studied the Markov process {(Z γ , U γ ); γ ∈ N}. As an application they considered a finite set of words W of equal length which are produced under the semi-Markovian hypothesis and focused on the waiting time for the first word occurrence from the set W in the semi-Markov sequence of letters.…”

Exact MLE and asymptotic properties for nonparametric semi-Markov models

“…The associated MC can play an important role in the understanding of the SMC. Actually, it can be useful, on one hand, to study the probabilistic behavior and limit theorems for SMCs (see, for example, Stenflo (1996) and Chryssaphinou et al (2008)), and on the other hand, it can be used to develop statistical inference for the SMMs. The goal of this paper is to develop the latter idea and study MLE for this class of models.…”