Journal of Theoretical Probability

, Volume 8, Issue 3, pp 571–599 | Cite as

Return probabilities for random walk on a half-line

  • Steven P. Lalley


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 Σxpx>0, the transition probabilities satisfyqn(x, y)∼CxyR−nn−3/2 asn→∞, and when Σxpx=0,qn(x, y)∼Cxyn−1/2.

Key Words

Irreducible Markov chain random walk with reflecting zone 


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Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Steven P. Lalley
    • 1
  1. 1.Purdue UniversityWest Lafayette

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