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

Abstract

The longest stretch L(n) of consecutive heads in n independent and identically distributed 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∕pn and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).