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The A.S. Limit Distribution of the Longest Head Run

Published online by Cambridge University Press:  20 November 2018

Tamás F. Móri
Tamás F. Móri*
Affiliation:
Department of Probability Theory and Statistics Eotvos Lor and University H-J088 Budapest Hungary
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Abstract

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It is well known that the length Zn of the longest head run observed in n tosses with a fair coin is approximately equal to log2n with a stochastically bounded remainder term. Though — log2n does not converge in law, in the present paper it is shown to have almost sure limit distribution in the sense of the a. s. central limit theorem having been studied recently. The results are formulated and proved in a general setup covering other interesting problems connected with patterns and runs such as the longest monotone block or the longest tube of a random walk.

Keywords

60C0560F15runspatternswaiting timesalmost sure central limit theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1245 - 1262
Copyright
Copyright © Canadian Mathematical Society 1993

References

[B] Bellman, R., Introduction to Matrix Analysis, McGraw-Hill, New York, 1960.Google Scholar
[BD] Berkes, I. and Dehling, H., Some limit theorems in log density , Ann. Probabl., to appear.Google Scholar
[BDM] Berkes, I., Dehling, H. and Mori, T.G., Counterexamples related to the a. s. central limit theorem, Studia Sci. Math. Hungar. 26(1991), 153–164.Google Scholar
[BGT] Bingham, N.H., Goldie, C.M. and Teugels, J.L., Regular Variation, Encyclopedia of Mathematics 27, Cambridge University Press, 1987.Google Scholar
[CsF] Csáki, E. and Foldes, A., The narrowest tube of a recurrent random walk, Z. Wahrsch. Verw. Geb. 66(1984), 387–403.Google Scholar
[CsFK] Csáki, E., Foldes, A. and Komlos, J., Limit theorems for Erdôs-Rényi type problems, Studia Sci. Math. Hungar. 22(1987), 321–332.Google Scholar
[CsR] Csorgő, M. and Révesz, P., Strong Approximation in Probability and Statistics, Akadémiai Kiadô, Budapest, 1981.Google Scholar
[ER] Erdős, R. and Révész, R., On the length of the longest head run. In: Topics in Information Theory, Colloquia Math. Soc. J. Bolyai 16, Keszthely, Hungary, (1975), 219–228.Google Scholar
[F] Fôldes, A., The limit distribution of the length of the longest head-run, Period. Math. Hungar. 10(1979), 301–310.Google Scholar
[GSW] Gordon, L., Schilling, M.F. and Waterman, M.S., An extreme value theory for long head runs, Probab. Th. Rel. Fields 72(1986), 279–287.Google Scholar
[M85] Móri, T. F., Large deviation results for waiting times in repeated experiments, Acta Math. Hungar. 45(1985),213-221.Google Scholar
[M92] Móri, T. F, On the strong law of large numbers for logarithmically weighted sums , Annales Univ. Sci. Budapest, R. Eotvos Nom., Sec. Math., (1992), to appear.Google Scholar
[R] Révész, P., Three problems on the length of increasing runs, Stoch. Proc. Appl. 15(1983), 169–179.Google Scholar
[SI Stout, W.F., Almost Sure Convergence, Academic Press, New York, 1974.Google Scholar