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An extreme value theory for long head runs
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  • Published: June 1986

An extreme value theory for long head runs

  • Louis Gordon1,
  • Mark F. Schilling1 &
  • Michael S. Waterman1 

Probability Theory and Related Fields volume 72, pages 279–287 (1986)Cite this article

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Summary

For an infinite sequence of independent coin tosses withP(Heads)=p∈(0,1), the longest run of consecutive heads in the firstn tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn(1-p) i.i.d. exponential random variables. These results are extended to the case of the longest head run interrupted byk tails. The mean length of this run is shown to be log(n)+klog(n)+(k+1)log(1−p)−log(k!)+k+γ/λ−1/2+ r1(n)+ o(1) where log=log1/p , γ=0.577 ... is the Euler-Mascheroni constant, λ=ln(1/p), andr 1(n) is small. The variance is π2/6λ2+1/12 +r 2(n)+ o(1), wherer 2(n) is again small. Upper and lower class results for these run lengths are also obtained and extensions discussed.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, 90089-1113, Los Angeles, California, USA

    Louis Gordon, Mark F. Schilling & Michael S. Waterman

Authors
  1. Louis Gordon
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  2. Mark F. Schilling
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  3. Michael S. Waterman
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This work was supported by a grant from the System Development Foundation

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Gordon, L., Schilling, M.F. & Waterman, M.S. An extreme value theory for long head runs. Probab. Th. Rel. Fields 72, 279–287 (1986). https://doi.org/10.1007/BF00699107

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  • Received: 15 April 1984

  • Revised: 02 September 1985

  • Issue Date: June 1986

  • DOI: https://doi.org/10.1007/BF00699107

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Class Result
  • Lower Class
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