CENTRAL LIMIT-THEOREMS FOR RANDOM-WALKS ON NO THAT ARE ASSOCIATED WITH ORTHOGONAL POLYNOMIALS

Central limit theorems are proved for Markov chains on the nonnegative integers that are homogeneous with respect to a sequence of orthogonal polynomials where the 3-term recurrence formula that defines the orthogonal polynomials has to satisfy some conditions. In particular, from the rate of convergence of the coefficients of the 3-term recurrence relation we get an estimation for the rate of convergence in the central limit theorems. The central limit theorems are applied to certain polynomial hypergroups, to birth and death random walks, and to isotropic random walks on infinite distance-transitive graphs and on certain finitely generated semigroups. © 1990.