CHARACTERIZATION OF MARKOV-BERNOULLI GEOMETRIC DISTRIBUTION RELATED TO RANDOM SUMS

The Markov-Bernoulli geometric distribution is obta ined when a generalization, as a Markov process, of  the independent Bernoulli sequence of random variab les is introduced by considering the success probability changes with respect to the Markov chai n. The resulting model is called the MarkovBernoulli model and it has a wide variety of applic at on fields. In this study, some characterizations are given concerning the Markov-Bernoulli geometric dis tribution as the distribution of the summation inde x of independent randomly truncated non-negative inte ger valued random variables. The achieved results generalize the corresponding characterizations conc erning the usual geometric distribution.


INTRODUCTION
The Markov-Bernoulli Geometric (MBG) distribution has been obtained by Anis and Gharib (1982) in a study of Markov-Bernoulli sequence of random variables (rv's) introduced by Edwards (1960), who generalized the (usual) independent Bernoulli sequence of rv's by considering the success probability changes with respect to a Markov chain. The resulting model is called the Markov-Bernoulli Model (MBM) or the Markov modulated Bernoulli process (Ozekici, 1997). Many researchers have been studied the MBM from the various aspects of probability, statistics and their applications, in particular the classical problems related to the usual Bernoulli model (Anis and Gharib, 1982;Gharib and Yehia, 1987;Inal, 1987;Yehia and Gharib, 1993;Ozekici, 1997;Ozekici and Soyer, 2003;Arvidsson and Francke, 2007;Omey et al., 2008;Maillart et al., 2008;Pacheco et al., 2009;Cekanavicius and Vellaisamy, 2010;Minkova and Omey, 2011). Further, due to the fact that the MBM operates in a random environment depicted by a Markov chain so that the probability of success at each trial depends on the state of the environment, this model represents an interesting application of stochastic processes and thus used by numerous authors in, stochastic modeling (Switzer, 1967;1969;Pedler, 1980;Satheesh et al., 2002;Özekici and Soyer, 2003;Xekalaki and Panaretos, 2004;Arvidsson and Francke, 2007;Nan et al., 2008;Pacheco et al., 2009;Doubleday and Esunge, 2011;Pires and Diniz, 2012).

JMSS
The sequence {X i } with the transition matrix (1.1) and the above initial distribution is called the MBM. If E i , i = 0, 1 are the states of the Markov system given by (1.1), then the parameter ρ which is usually called the persistence indicator of E 0 , is the correlation coefficient between X i and X i+1 , i = 1, 2,… (Anis and Gharib, 1982).
If N is the number of transitions for the system defined by (1.1) to be in E 1 for the first time then N has the MBG distribution given by Equation (1.2): The MBG distribution (1.2) will be denoted by MBG(α, ρ) and we shall write N∼MBG(α, ρ). Let P U (s) = E(s U ) be the probability generating function (pgf) of an integer valued random variable U, |s|≤1.
The pgf of N is given, using (1.2), by Equation (1.3): Few authors have treated the characterization problem of the MBG distribution (Yehia and Gharib, 1993;Minkova and Omey, 2011).
In this study some new characterizations are given for the MBG distribution by considering it as the distribution of the summation index of a random sum of randomly truncated non-negative integer valued rv's. The scheme of geometric random sum of randomly truncated rv's is important in reliability theory, especially in the problem of optimal total processing time with checkpoints . The achieved results generalize those given in .

Remark 1
If the sequence {X n , n≥1} follows the MBM (1.1), where the event {(X i ≥ Y i )} indicates the state E 0 and the event {(X i <Y i )} indicates the state E 1 , with the initial distribution: Then the rv N (the smallest number of transitions for the system to be in state E 1 for the first time) wouldhave the MBG distribution defined by (1.2) (Anis and Gharib, 1982) Equation (2.1): Define: where, for convenience Y 0 = 0. The rv Z is the random sum of truncated rv's . It is known that the pgf P Z (s) of Z is given by  Equation (2.2 to 2.5): where: Corollary 1 The result is immediate since E(Z) = P′z (1).

CHARACTERIZATIONS OF THE MBG DISTRIBUTION
Consider the random sum given by (2.1).

Theorem 1
Let Y 1 , have a MBG distribution with some parameters α∈(0, 1) and ρ∈[0, 1] and let X 1 satisfy P(X 1 = 0)<1. Then Z has the same distribution as X 1 1 ( ) d Z X = if and only if X 1 has a MBG distribution.
This completes the proof of Theorem 1.

Remark 2
It follows from Theorem 1 that when X 1 has the MBG distribution then the random sum Z and the summands have distributions of the same type and in this case the summands are called N-sum stable (Satheesh et al., 2002). This result is valid, also, as a consequence of the fact that geometric random sums are stable in the same sense.
Another characterization for the MBG distribution can be obtained in terms of the expected values of Z and X 1 .
This completes the proof of Theorem 2.
The following theorem expresses the relation between the distribution of Z and the distribution of the truncating process {Y n }.
The "only if" part of the proof follows directly by applying Theorem 1.
This completes the proof of Theorem 3.

Remark 3
The results of  follow as special cases from our corresponding results when the MBM (1.1) reduces to the independence case by putting the correlation parameter ρ = 0.

CONCLUSION
In this study three characterizations for the Markov-Bernoulli geometric distribution are proved. These results extend the corresponding characterizations of the geometric distribution. Further, the achieved results have a direct relevance to the stability problem of random sums of random variables. Moreover, the given characterization theorems will be useful, in understanding the missing link between the mathematical structure of Markov-Bernoulli geometric distribution and the actual behavior of some real world random phenomena.