S. N. Ethier
Consider the random walk on the set of nonnegative integers that takes two steps to the left (just one step from state 1) with probability p∈[1/3,1) and one step to the right with probability 1−p. State 0 is absorbing and the initial state is a fixed positive integer j0. Here we find the distribution of the absorption time. The absorption time is the duration of (or the number of coups in) the well-known Labouchere betting system. As a consequence of this, we obtain in the fair case (p=1/2) the asymptotic behavior of the Labouchere bettor’s conditional expected deficit after n coups, given that the system has not yet been completed.
References
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Mathematical Reviews (MathSciNet):
MR590362
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Mathematical Reviews (MathSciNet):
MR67418
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