A Markov-binomial distribution

Abstract

Let {Xi, i ≥ 1} denote a sequence of {0, 1}-variables and suppose that the sequence forms a Markov Chain. In the paper we study the number of successes Sn = X1 + X2 + • • • + Xn and we study the number of experiments Y (r) up to the r-th success. In the i.i.d. case Sn has a binomial distribution and Y (r) has a negative binomial distribution and the asymptotic behaviour is well known. In the more general Markov chain case, we prove a central limit theorem for Sn and provide conditions under which the distribution of Sn can be approximated by a Poisson-type of distribution. We also completely characterize Y (r) and show that Y (r) can be interpreted as the sum of r independent r.v. related to a geometric distribution.

Delay-exponent of decode-forward streaming

“…The generalised binomial distribution [11] models the first property, where the current event is equally dependent on all past events. The Markov binomial distribution [12], on the other extreme, only considers dependency between consecutive events.…”

“…−k converges sufficiently fast to p k then the exponent of the terms within the summation in (12) approaches that achieved by fully source-relay cooperative…”

“…A summary of the characteristics of the process b i and the sum η(t, ) is given in [12]. Since Pr{b i = 1} = p k , we have that a 0,1 a 0,1 + a 1,0 = p k .…”

Characterization of Markov-Bernoulli Geometric Distribution Related to Random Sums

“…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).…”

Moderate and Large Deviation Estimate for the Markov-Binomial Distribution

“…Wang [16] and Omey et al [14] studied the asymptotic distribution of the Markov-binomial distribution. Cekanavičius and Mikalauskas [4] obtained the large deviations estimate for the Poisson approximation of the Markov-binomial distribution S n .Čekanavičius and Roos [5] considered the binomial approximation of the Markov binomial distribution and gave the sharp estimates for the total variation and local norms.…”