Limit theorems for the number of occurrences of consecutive k successes in n Markovian trials

Abstract

We present a method of deriving the limiting distributions of the number of occurrences of success (S) runs of length k for all types of runs under the Markovian structure with stationary transition probabilities. In particular, we consider the following four bestknown types. 1. A string of S of exact length k preceded and followed by an F, except the first run which may not be preceded by an F, or the last run which may not be followed by an F. 2. A string of S of length k or more. 3. A string of S of exact length k, where recounting starts immediately after a run occurs. 4. A string of S of exact length k, allowing overlapping runs. It is shown that the limits are convolutions of two or more distributions with one of them being either Poisson or compound Poisson, depending on the type of runs in question. The completely stationary Markov case and the i.i.d. case are also treated.