Downloaded By: [Limnios, Nikolaos] At: 16:01 16 February 2008
Communications in Statistics—Theory and Methods, 37: 1306–1322, 2008
Copyright © Taylor & Francis Group, LLC
ISSN: 0361-0926 print/1532-415X online
DOI: 10.1080/03610920701713328
Applied Probability
On Discrete Time Semi-Markov Chains
and Applications in Words Occurrences
OURANIA CHRYSSAPHINOU
1
,
MARGARITA KARALIOPOULOU
1
,
AND NIKOLAOS LIMNIOS
2
1
Department of Mathematics, University of Athens, Greece
2
Laboratoire de Mathematiques Appliquées,
Université de Technologie de Compiègne, France
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 = 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.
Keywords Backward recurrence times; Discrete time semi-Markov process; First
occurrence; Waiting time; Words.
Mathematics Subject Classification Primary 60K15; Secondary 60K20.
1. Introduction
Let us consider a sequence of outcomes Z
∈ generated by a semi-Markov
chain with finite state space =
1
2 ≤ <
We study the joint distribution of the semi-Markov process Z
∈
and its corresponding backward recurrence time U
∈ The Markov chain
Z
U
∈ is the basic process for our study and plays an important role in
the theory and applications of discrete time semi-Markov processes.
Received June 15, 2006; Accepted July 31, 2007
Address correspondence to Nikolaos Limnios, Université de Technologie de Compiègne,
Laboratoire de Mathematiques Appliquées Centre de Recherches de Royallieu, BP 20529,
Compiègne Cedex 60205, France; E-mail: Nikolaos.Limnios@utc.fr
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