Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials. (English) Zbl 1351.60030

Konstantopoulos, Takis; Liu, Zhenxia; Yang, Xiangfeng

J. Appl. Probab. 53, No. 3, 747-764 (2016).

Summary: 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 \(\log_{1/p}n\) and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of \(L(n)\).

MSC:

60F10	Large deviations
44A10	Laplace transform
60G50	Sums of independent random variables; random walks
60G70	Extreme value theory; extremal stochastic processes

Keywords:

large deviation principle; rate function; Fenchel-Legendre transform; Laplace transform; moment generating function; run; longest run; Bernoulli trial; confidence interval