Published online by Cambridge University Press: 20 November 2018
It is well known that the length Zn of the longest head run observed in n tosses with a fair coin is approximately equal to log2 n with a stochastically bounded remainder term. Though — log2 n does not converge in law, in the present paper it is shown to have almost sure limit distribution in the sense of the a. s. central limit theorem having been studied recently. The results are formulated and proved in a general setup covering other interesting problems connected with patterns and runs such as the longest monotone block or the longest tube of a random walk.
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