Absorption Time Distribution for an Asymmetric Random Walk

Abstract

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.