A Conditioned Random Walk with Applications

摘要

A random walk on the set of integers {0, 1, 2,..., N} with absorbing barriers at 0 and N is considered. Transition times from, the points r (0 < r < N) are random variables with finite expectations that depend only on r. This report presents mean time to absorption atnN, Dr, with the condition that the walk starts at r and is absorbed at N. A probability mea sure is put on the random walk histories. A difference equation with boundary values for D is derived. It is solved in the special case of a birth and death process with linear growth. Higher moments of the time to absorption are discussed. This problem is motivated by two mathematical models in biology, which are also discussed.