Random Walks and Convolution on Groups and Semigroups

Summary

A classical result of P. Lévy on random rotations on the circle is derived in the first section. The distribution of the product of two random factors is given by the convolution of their distributions. Convolution of measures on a compact semigroup is introduced in section 2 and some of its properties are derived. The right and left random walks induced by a measure on a semigroup are introduced. In the next section, it is shown that averages of the sequence of convolutions of a regular measure with itself on a compact semigroup converge to an idempotent measure. This provides a constructive way of deriving the Haar measure on a compact group. This method of obtaining the Haar measure on a compact group appears to be due to Ito and Kawada. The algebraic structure of compact semigroups is dealt with in section 4. The minimal right (left) and two-sided ideals in a compact semigroup are characterized. This is extremely useful in determining the form of idempotent probability measures on a compact semigroup. The ideas and results of this section are illustrated in the case of the semigroup of n × n (n finite) stochastic matrices. The last section of the chapter presents conditions under which the unaveraged sequence of convolutions of a regular probability measure with itself converge. Conditions under which stationary right and left random walks on a compact group or semigroup are ergodic or mixing are determined.