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We study the length of $T$-contaminated runs of heads in the well-known cointossing experiment. A $T$-contaminated run of heads is a sequence ofconsecutive heads interrupted by $T$ tails. For $T=1$ and $T=2$ we find theasymptotic distribution for the first hitting time of the $T$ contaminated runof heads having length $m$; furthermore, we obtain a limit theorem for thelength of the longest $T$-contaminated head run. We prove that the rate of theapproximation of our accompanying distribution for the length of the longest$T$-contaminated head run is considerably better than the previous ones. Forthe proof we use a powerful lemma by Cs\'aki, F\"oldes and Koml\'os.

Convergence rate for the longest T-contaminated runs of heads. Paper with detailed proofs