Chryssaphinou, Ourania; Karaliopoulou, Margarita; Limnios, Nikolaos On discrete time semi-Markov and applications in words occurences. (English) Zbl 1167.60357 Commun. Stat., Theory Methods 37, No. 8, 1306-1322 (2008). Summary: Let a discrete time semi-Markov process \(\{Z_\gamma;\gamma\in\mathbb N\}\) with finite state space an alphabet \(\Omega\). Defining the process \(\;U_\gamma; \gamma\in\mathbb N\;\) to be the backward recurrence time of the process \(\{\mathbb Z_\gamma; \gamma\in\mathbb N\}\), we study the Markov process \(\{(Z_\gamma, U_\gamma); \gamma\in\mathbb N\}\). 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 \(\{(\overline Z_\gamma, \overline U_\gamma); \gamma\in\mathbb N\}\) and we study its basic properties. As an application we consider a finite set of words \(W = \{w_1, w_2,\dots, w_\nu\}\) 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. Cited in 11 Documents MSC: 60K15 Markov renewal processes, semi-Markov processes 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.) Keywords:backward recurrence times; discrete time semi-Markov process; first occurrence; waiting time; words PDFBibTeX XMLCite \textit{O. Chryssaphinou} et al., Commun. Stat., Theory Methods 37, No. 8, 1306--1322 (2008; Zbl 1167.60357) Full Text: DOI References: [1] DOI: 10.1239/jap/996986759 · Zbl 0985.60072 [2] DOI: 10.1081/STA-200037923 · Zbl 1089.60525 [3] Biggins J. D., Adv. Appl. Prob. 19 pp 521– (1987) · Zbl 0629.60100 [4] Blom G., J. Appl. Prob. 19 pp 518– (1982) · Zbl 0493.60071 [5] DOI: 10.2307/1426216 · Zbl 0212.49601 [6] Chryssaphinou , O. , Karaliopoulou , M. , Limnios , N. ( 2005 ). On occurrences of words under Markovian hypothesis . In Proc. of the 11th International Symposium on Applied Stochastic Models and Data Analysis (ASMDA 2005) . France : Brest , pp. 1133 – 1140 . [7] Chryssaphinou O., Theory Probab. Appl. 35 pp 167– (1990) [8] Chryssaphinou O., Runs and Patterns in Probability pp 231– (1990) [9] Feller W., An Introduction to Probability Theory and Its Applications. (1968) · Zbl 0155.23101 [10] DOI: 10.1016/0097-3165(81)90005-4 · Zbl 0454.68109 [11] Howard R., Dynamic Probabilistic Systems (1971) · Zbl 0227.90031 [12] DOI: 10.1137/1.9780898719734 · Zbl 0922.60001 [13] Limnios N., Semi-Markov Processes and Reliability. (2001) · Zbl 0990.60004 [14] DOI: 10.1214/aoms/1177704863 · Zbl 0267.60089 [15] Pyke R., Ann. Math. Statist. 32 pp 1243– (1961) · Zbl 0201.49901 [16] DOI: 10.1214/aoms/1177700397 · Zbl 0134.34602 [17] Robin S., J. Appl. Prob. 36 pp 179– (1999) · Zbl 0945.60008 [18] Stefanov V. T., J. Appl. Prob. 37 pp 756– (2000) · Zbl 0969.60021 [19] Stefanov V. T., J. Appl. Prob. 40 pp 881– (2003) · Zbl 1054.60022 [20] Yackel J., Trans. Amer. Math. Soc. 123 pp 402– (1969) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.