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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: takiskonst@gmail.com
** 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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References

[1] Balakrishnan, N. and Koutras, M. V.(2002).Runs and Scans with Applications.John Wiley,New York.Google Scholar
[2] Clopper, C. J. and Pearson, E. S. (1934).The use of confidence or fiducial limits illustrated in the case of the binomial.Biometrika 26,404–413.Google Scholar
[3] Dembo, A. and Zeitouni, O.(2010).Large Deviations Techniques and Applications, 2nd edn.Springer,Berlin.CrossRefGoogle Scholar
[4] Erdős, P. and Rényi, A. (1970).On a new law of large numbers.J. Analyse Math. 33,103–111.CrossRefGoogle Scholar
[5] Erdős, P. and Révész, P. (1977).On the length of the longest head-run. InTopics in Information Theory (Colloq. Math. Soc. Janos Bolyai 16),North-Holland,Amsterdam. pp. 219–228.Google Scholar
[6] Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn.John Wiley,New York.Google Scholar
[7] Fu, J. C.,Wang, L. and Lou, W. Y. W. (2003).On exact and large deviation approximations for the distribution of the longest run in a sequence of two-state Markov dependent trials.J. Appl. Prob. 40,346–360.Google Scholar
[8] Holst, L. and Konstantopoulos, T. (2015).Runs in coin tossing: a general approach for deriving distributions for functionals.J. Appl. Prob. 52,752–770.Google Scholar
[9] Mao, Y.-H.,Wang, F. and Wu, X.-Y. (2015).Large deviation behavior for the longest head run in an IID Bernoulli sequence.J. Theoret. Prob. 28,259–268.Google Scholar
[10] Novak, S. Y. (1992).Longest runs in a sequence of m-dependent random variables.Prob. Theory Relat. Fields 91,269–281.Google Scholar
[11] Philippou, A. N. and Makri, F. S. (1985).Longest success runs and Fibonacci-type polynomials. Fibonacci Quart. 23,338–346.Google Scholar
[12] Rényi, A. (1970).Foundations of Probability.Holden-Day, San Francisco, CA.Google Scholar
[13] Sandmann, K. and Schönbucher, P. (2002).Advances in Finance and Stochastics: Essays in Honour of DieterSondermann.Springer,Berlin.Google Scholar
[14] Schilling, M. F. (1990).The longest run of heads.College Math. J. 21,196–207.CrossRefGoogle Scholar
[15] Von Mises, R. (1921).Zur Theorie der Iterationen.Z. Angew. Math. Mech. 1,298–307.Google Scholar
[16] Von Mises, R. (1981).Probability, Statistics, and Truth, 2nd edn.Dover,New York.Google Scholar
[17] Wentzell, A. D. (1990).Limit Theorems on Large Deviations for Markov Stochastic Processes. Kluwer,Dordrecht.Google Scholar
[18] Wilson, E. B. (1927).Probable inference, the law of succession, and statistical inference.J. Amer. Statist. Assoc. 22,209–212.Google Scholar
[19] Zhang, X. Y. and Wu, X. Y. (2013).Some results associated with the longest run in a strongly ergodic Markov chain.Acta Math. Sin. (Engl. Ser.) 29,1939–1948.Google Scholar