LIMIT-THEOREMS FOR RANDOM-WALKS CONDITIONED TO STAY POSITIVE

Let {S(n)} be a random walk on the integers with negative drift, and let A(n) = {S(k) greater-than-or-equal-to 0, 1 less-than-or-equal-to k less-than-or-equal-to n} and A = A(infinity). Conditioning on A is troublesome because P(A) = 0 and there is no natural sigma-field of events "like" A. A natural definition of P(B\A) is lim(n --> infinity) P(B\A(n)). The main result here shows that this definition makes sense, at least for a large class of events B: The finite-dimensional conditional distributions for the process {S(k)}k greater-than-or-equal-to 0 given A(n) converge strongly to the finite-dimensional distributions for a measure Q. This distribution Q is identified as the distribution for a stationary Markov chain on {0, 1,...}.