Random walk polynomials and random walk measures

Random walk polynomials and random walk measures play a prominent role in the analysis of a class of Markov chains called random walks. Without any reference to random walks, however, a random walk polynomial sequence can be defined (and will be defined in this paper) as a polynomial sequence{Pn(x)} which is orthogonal with respect to a measure on [-1, 1] and which is such that the parameters (alfa)n in the recurrence relations Pn=1(x)=(x(alfa)n)Pn(x)-ßnPn-1(x) are nonnegative. Any measure with respect to which a random walk polynomial sequence is orthogonal is a random walk measure. We collect some properties of random walk measures and polynomials, and use these findings to obtain a limit theorem for random walk measures which is of interest in the study of random walks. We conclude with a conjecture on random walk measures involving