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

Part of: Integral transforms, operational calculus Limit theorems Stochastic processes

Published online by Cambridge University Press:  24 October 2016

Takis Konstantopoulos ,
Zhenxia Liu  and
Xiangfeng Yang
Takis Konstantopoulos*
Affiliation:
Uppsala University
Zhenxia Liu*
Affiliation:
Linköping University
Xiangfeng Yang*
Affiliation:
Linköping University
*
* Postal address: Department of Mathematics, Uppsala University, SE-751 06 Uppsala, Sweden. Email address: [email protected]
** Postal address: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden.
** Postal address: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden.
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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 log1∕p n and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

Keywords

Large deviation principlerate functionFenchel–Legendre transformLaplace transformmoment generating functionrunlongest runBernoulli trialconfidence interval

MSC classification

Primary: 60F10: Large deviations
Secondary: 44A10: Laplace transform 60G50: Sums of independent random variables; random walks 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 747 - 764
Copyright
Copyright © Applied Probability Trust 2016 

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