Some New Characterizations of Markov-Bernoulli Geometric Distribution Related to Random Sums

The Markov-Bernoulli geometric distribution is obtained when a generalization, as a Markov process, of the independent Bernoulli sequence of random variables, is introduced. In this paper, new characterizations of the Markov-Bernoulli geometric distribution, as the distribution of the summation index of randomly truncated nonnegative integer valued random variables, are given in terms of moment relations of the sum and summands. The achieved results generalize the corresponding characterizations concerning the usual geometric distribution.


Introduction
Many basic counting distributions are defined on a se-quence of independent identically distributed (iid) Bernoulli random variables (rv's).Many other distributions are defined by compounding and mixing.Another way of obtaining new discrete distributions is to define the counting distributions related to some Markov chain.Assuming some dependency in the sequence of Bernoulli rv's gives an additional parameter by which the Bernoulli model could be a more realistic model in practice.Edwards (1960) proposed a generalization of the sequence of independent Bernoulli trials by considering the success probability evolves over time according to a Markov chain.In other words, let X 1 , X 2 , . . .be a sequence of Bernoulli rv's with the following one step transition probabilities matrix and initial distribution: where p ∈ [0, 1] and ρ ∈ [0, 1] .The additional parameter ρ is the correlation coefficient between X i and X i+1 , and is, also, called the persistent indicator of the initial state of the system (1.1) (Wang, 1981).
The Markov-Bernoulli geometric (MBG) distribution has been obtained by Anis and Gharib (1982) in an earlier detailed study of MBM.If E i ; i = 0; 1 are the states of the system (1.1) and if N is a rv representing the number of trials necessary for the system to be in state E 1 for the first time, then the probability mass function (pmf) of the rv N is given by: The distribution (1.2) is the MBG distribution.It represents a generalization of the usual geometric distribution (ρ = 0).In the past works on the MBM some characterizations for the MBG distribution (1.2) are achieved (Minkova & Omey, 2011;Yehia & Gharib, 1993).The random sum of independent identically distributed (iid) nonnegative rv's; where the summation index is a geometric rv; is called a geometric compounding of the underlying rv's.Such a compounding mechanism is closely related to the rarefaction (or the thinning version) of a renewal process and has practical applications to the traffic theory, reliability and ecology problems involving rare events (Gnedenko & Korolev, 1996).These applications motivated many researchers to characterize such random sums, see e.g.Gharib et al. (2003), Khalil et al. (1991), Milne and Yeo (1989) and the references therein.
In this paper, we are interested in characterizing the random sum: introduced by Khalil et al. (1991).
The rv Z represents the truncated sum until the moment where for the first time the process {Y n ; n ≥ 1} has greater jump than the corresponding jump of the process {X n ; n ≥ 1}.The sequence {Y n } is called the truncating process (Gharib et al., 2014).In queuing systems with unreliable server, Z can be interpreted as the total time duration of the unreliable server until the successful finish of the service if the corresponding duration without breakdowns is previously known (Dimitrov et al., 1991).The random sum (1.3) also arises in some models such as unreliable computing systems, multiclients computer service, data transmissions, constructing data and so on, where in a random environment some undesirable events (breakdowns, interruptions, etc.) arise and lead to process interruption or to incorrect final results (Duda, 1983;Koren et al., 1986).
Throughout the paper, we write P U (s) = E(s U ); |s| ≤ 1, to denote the probability generating function (pgf) of an integer-valued rv U, W ∼ MBG(α, c) to denote a MBG distributed rv W with pmf: and U d = V to denote a rv U having the same distribution as a rv V. Let I(A) be the indicator of a set A and put (1.5) The pgf of Z in (1.3) is given by: Khalil et al., 1991).
In the following, we list some results obtained by Gharib et al. (2014) which are of direct relevance to the development of the results of the present paper.Consider the random sum Z defined by (1.3).
Theorem 1.1 Assume Y 1 ∼ MBG(α, ρ), for some parameters α ∈ (0; 1) and ρ ∈ [0, 1], and X 1 is an arbitrarily non-negative integer-valued rv with p 0 = P(X The organization of this paper is as follows.In section 2, we derive certain differential equations associated with the random sum (1.3).Also, the unique solutions of these differential equations are given under certain initial conditions.In section 3, we provide three new characterizations of the MBG distribution related to the random sum (1.3) based on certain relations between the moments of Z and/or X 1 .Finally some concluding remarks are given.

Preliminaries
The following lemma derives the main differential equations that will be used in the proofs of the new characterizations of the MBG distribution.
(iii) Proof of this part is similar to that of (i) and (ii).
Remark 2.2 The differential equations derived in Lemma 2.1 has the following general form: where a k , a k−1 , ..., a 0 are constants, φ (k) (t) denotes the k th derivative of φ(t), and g(t) is a non-zero function of t.This differential equation is known as the k th order nonhomogeneous Euler differential equation (Zwillinger, 1992, p. 235).

Characterization Results
The first characterization of the MBG distribution related to the random sum (1.3) is expressed in terms of the first two moments of Z.
The second characterization of the MBG distribution related to the random sum (1.3) is given in terms of the first three moments of Z and the first moment of X 1 .
The third characterization of the Markov-Bernoulli geometric distribution related to the random sum (1.3) is expressed in terms of the first four moments of Z and the first two moments of X 1 .

Concluding Remarks
1) From the "if" part of Theorems 3.1-3.3and according to Theorem 1.1, It follows 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.
2) Each of the three nonhomogeneous differential equations used in the characterizations of the Markov-Bernoulli geometric distribution given in this paper has a linear nonhomogeneous term.The solution associated with each of these differential equations under certain initial condition(s) is essentially its particular solution.
3) We may note that the condition of Theorem 3.1 involves only the moments of Z while the conditions of Theorem 3.2 and Theorem 3.3 involve both the moments of Z and X 1 : Indeed, this is due to the nature of the initial conditions used in the sufficiency part of the proofs of these theorems.