A MARKOV-BINOMIAL DISTRIBUTION

Let ${X_{i},igeq 1}$ denote a sequence of $left{ 0,1 ight} $%-variables and suppose that the sequence forms a {sc Markov} Chain. In the paperwe study the number of successes $S_{n}=X_{1}+X_{2}+cdots+X_{n}$ and we studythe number of experiments $Y(r)$ up to the $r$-$th$ success. In the i.i.d.case $S_{n}$ has a binomial distribution and $Y(r)$ has a negative binomialdistribution and the asymptotic behaviour is well known. In the more general{sc Markov} chain case, we prove a central limit theorem for $S_{n}$ and provideconditions under which the distribution of $S_{n}$ can be approximated by a{sc Poisson}-type of distribution. We also completely characterize $Y(r)$ andshow that $Y(r)$ can be interpreted as the sum of $r$ independent r.v.related to a geometric distribution.