Central limit theorems are proved for random walks on the non-negative integers where the transition probabilities are homogeneous with respect to a sequence of orthogonal polynomials. Assuming some restrictions concerning the three-term recursion formula of these polynomials, one gets a Rayleigh distribution as limit distribution where bounds of the order of convergence can be computed explicitly. These central limit theorems are applied to generalized birth and death random walks and random walks on polynomial hypergroups. Finally some examples of polynomial hypergroups are discussed in view of the limit theorems above.

The process generated by the crossings of a fixed level, u, by the process Pn(t) is considered, where

and the Xi(t) are identical, independent, separable, stationary, zero mean, Gaussian processes. A simple formula is obtained for the expected number of upcrossings in a given time interval, sufficient conditions are given for the upcrossings process to tend to a Poisson process as u→∞, and it is shown that under suitable scaling the distribution of the length of an excursion of Pn(t) above u tends to a Rayleigh distribution as u→ ∞.