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 {P(n)(x)) which is orthogonal with respect to a measure on [- 1, 1] and which is such that the parameters alpha(n) in the recurrence relations P(n+1)(x)=(x - alpha(n))P(n)(x) - beta(n)P(n-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 Christoffel functions.