Return probabilities for random walk on a half-line

A random walk with reflecting zone on the nonnegative integers is a Markov chain whose transition probabilitiesq(x, y) are those of a random walk (i.e.,q(x, y)=p(y–x)) outside a finite set {0, 1, 2,...,K}, and such that the distributionq(x,·) stochastically dominatesp(·–x) for everyx{0, 1, 2,..., K}. Under mild hypotheses, it is proved that when xp_x>0, the transition probabilities satisfyqⁿ(x, y)C_xyR^–nn^–3/2 asn, and when xp_x=0,qⁿ(x, y)C_xyn^–1/2.Supported by National Science Foundation Grant DMS-9307855.