Gambler’s Ruin: A Random Walk on the Simplex

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

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Gambler's Ruin and the First Exit Position of Random Walk from Large Spheres

The Annals of Probability ◽

10.1214/aop/1176988608 ◽

1994 ◽

Vol 22 (3) ◽

pp. 1429-1472 ◽

Cited By ~ 11

Author(s):

Philip S. Griffin ◽

Terry R. McConnell

Keyword(s):

Random Walk ◽

Gambler’S Ruin

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Random walks with negative drift conditioned to stay positive

Journal of Applied Probability ◽

10.2307/3212557 ◽

1974 ◽

Vol 11 (4) ◽

pp. 742-751 ◽

Cited By ~ 29

Author(s):

Donald L. Iglehart

Keyword(s):

Random Walk ◽

Random Walks ◽

Random Variable ◽

Principal Result ◽

Collective Risk ◽

The Laplace Transform ◽

Negative Drift ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem ◽

Independent Identically Distributed

Let {Xk: k ≧ 1} be a sequence of independent, identically distributed random variables with EX1 = μ < 0. Form the random walk {Sn: n ≧ 0} by setting S0 = 0, Sn = X1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X1) that Sn, conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

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Random walks with negative drift conditioned to stay positive

Journal of Applied Probability ◽

10.1017/s0021900200118170 ◽

1974 ◽

Vol 11 (04) ◽

pp. 742-751 ◽

Cited By ~ 11

Author(s):

Donald L. Iglehart

Keyword(s):

Random Walk ◽

Random Walks ◽

Random Variable ◽

Principal Result ◽

Collective Risk ◽

The Laplace Transform ◽

Negative Drift ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem ◽

Independent Identically Distributed

Let {Xk : k ≧ 1} be a sequence of independent, identically distributed random variables with EX 1 = μ < 0. Form the random walk {Sn : n ≧ 0} by setting S 0 = 0, Sn = X 1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X 1) that Sn , conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

Download Full-text