Limit theorems for contaminated runs of heads

Abstract. The classical coin tossing experiment is studied. Limit theorems are obtained concerning the head-runs containing certain number of tails. It is proved that the limit of the number of those runs of length \(n\) which contain at most \(T\) tails is compound Poisson. Accompanying distributions are obtained for the length of the longest head-runs containing at most \(T\) tails. To this end a two parameter family of accompanying distributions is offered.

Key words and phrases. Coin tossing, head-run, contaminated run, limit distribution, Poisson distribution.