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Moments in the duration of play

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  • Bach, Eric

Abstract

We evaluate moments of the completion time in the classical duration of play problem, equivalent to a symmetric random walk with absorbing endpoints on t-n,h.,n starting from x. We show that the rth cumulant of the absorption time has the form Pr(n2) - Pr(x2) where Pr is a degree r polynomial, and evaluate Pr for r = 1,...,6. Measured in arithmetic operations, our algorithm to compute Pr has amortized cost comparable to the inversion of an r x r matrix. We obtain similar results for unequal initial stakes, and under conditioning on a win by one player. The conditional results settle some open questions due to Beyer.

Suggested Citation

  • Bach, Eric, 1997. "Moments in the duration of play," Statistics & Probability Letters, Elsevier, vol. 36(1), pages 1-7, November.
    • Handle: RePEc:eee:stapro:v:36:y:1997:i:1:p:1-7
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    1. Huckaby, Dale A. & Hubbard, Joseph B., 1983. "A random walk on a random channel with absorbing barriers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 122(3), pages 602-610.
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    Cited by:

    1. Tamás Lengyel, 2011. "Gambler’s ruin and winning a series by m games," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(1), pages 181-195, February.
    2. Anděl, Jiří & Hudecová, Šárka, 2012. "Variance of the game duration in the gambler’s ruin problem," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1750-1754.

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      Keywords

      Random walk Ruin problem;

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