Limit theorems for the number of occurrences of consecutiveksuccesses innMarkovian trials
Y. H. Wang, Shuixin Ji · Journal of Applied Probability · 1995
We present a method of deriving the limiting distributions of the number of occurrences of success(S)runs of lengthkfor all types of runs under the Markovian structure with stationary transition probabilities. In particular, we consider the following four bestknown types. 1. A string ofSof exact lengthkpreceded and followed by anF, except the first run which may not be preceded by anF, or the last run which may not be followed by anF. 2. A string ofSof lengthkor more. 3. A string ofSof exact lengthk, where recounting starts immediately after a run occurs. 4. A string ofSof exact lengthk, 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.