Limit theorems for random walks on discrete semigroups related to nonhomogeneous trees and Chebyshev polynomials (Q1122855)

The author introduces a family of discrete semigroups \(S_ N\) \((N=1,2,...)\), and sequences of independent, identically distributed random variables \(X_ 1,X_ 2,X_ 3,...,X_ n,..\). with values in \(S_ N\). The main results consist in some limit theorems for the product variables \(Y_ n=X_ 1X_ 2...X_ n.\) The author proves a general law of large numbers for \(| Y_ n|\), where \(| \cdot |\) denotes the length function on \(S_ N\). He also proves the central limit theorem for a large class of the above processes. A motivation for this paper is to suggest that random walks on certain nonhomogeneous trees could be studied as random walks on semigroups. The author points out that a class of random walks on \(S_ 1\) is equivalent to the random walks on hypergroups connected to the Chebyshev polynomials of the second kind.