On long runs of heads and tails (Q1314683)

In fair coin tossing, i.e. for an i.i.d. sequence of Bernoulli random variables with values \(H\) (head) and \(T\) (tail), both occurring with probability \(1/2\), let \(Z_ H(n)\) and \(Z_ T(n)\) denote the length of the longest head resp. tail run observed during the first \(n\) tosses. It is shown that the logarithmic density of the random sequence of those \(n\) for which the difference \(Z_ H(n)-Z_ T(n)\) falls into a fixed subset of the set of all integers exists with probability one, and the value of this density is given explicitly.