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How many random digits are required until given sequences are obtained?

Published online by Cambridge University Press:  14 July 2016

Gunnar Blom and
Daniel Thorburn
Gunnar Blom*
Affiliation:
University of Lund
Daniel Thorburn*
Affiliation:
University of Lund
*
∗ Postal address: Department of Mathematical Statistics, University of Lund, Box 725, S-220 07 Lund, Sweden.
∗∗ Present address: Department of Mathematical Statistics, University of Stockholm, Box 6701, S-113 85 Stockholm, Sweden.
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Abstract

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

Keywords

LEADING NUMBERSOVERLAPRENEWAL PROCESSSTOPPING SEQUENCE

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982, pp. 518 - 531
Copyright
Copyright © Applied Probability Trust 1982 

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References

Blom, G. (1982) On the mean number of random digits until a given sequence occurs. J. Appl. Prob. 19, 136–143.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications, 3rd edn. Wiley, New York.Google Scholar
Gardner, M. (1974) Mathematical games. Scientific American 231 (October), 120–125.CrossRefGoogle Scholar
Gerber, H. U. and Li, S. R. (1981) The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stoch. Proc. Appl. 11, 101–108.CrossRefGoogle Scholar
Guibas, L. J. and Odlyzko, A. M. (1981) String overlaps, pattern matching and non-transitive games. J. Combinatorial Theory A 30, 183–208.Google Scholar
Howard, R. A. (1971) Dynamic Probabilistic Systems, Vol. 1. Wiley, New York.Google Scholar
Li, S. R. (1980) A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Prob. 8, 1171–1176.CrossRefGoogle Scholar
Nielsen, P. T. (1973) On the expected duration of a search for a fixed pattern in random data. IEEE Trans. Inform. Theory 19, 702–704.Google Scholar
Råde, L. (1972) Thinning of Renewal Point Processes. A Flow Graph Study. Matematisk statistik AB, Göteborg.Google Scholar
Solov'Ev, A. D. (1966) A combinatorial identity and its application to the problem concerning the first occurrence of a rare event. Theory Prob. Appl. 11, 276–282.Google Scholar
Todhunter, I. (1865) A History of the Mathematical Theory of Probability. (Reprinted (1965) by Chelsea, New York.) Google Scholar
Whitworth, W. A. (1901) Choice and Chance. (Reprinted (1948) by Hafner, New York.) Google Scholar