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Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials

The longest stretch $L(n)$ of consecutive heads in $n$ i.i.d. coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of $L(n)$ and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of $L(n)$ near its nominal value $\log_{1/p} n$ and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of $L(n)$.