Weak convergence of convolution iterates of probability measures on topological semigroups

This dissertation considers the problem of weak convergence of convolution products of non-identical probability measures with supports in a topological semigroup. Most of the work in the area of convolution products has dealt with the case of products of identical probability measures. When the measures are not necessarily the same, the situation is more complex and the theory much less developed.

After providing the necessary definitions and previous results in the area in Chapter 1, the second chapter begins the examination of the problem, in locally compact and compact semigroups. The results of this chapter include determining an algebraic structure for the tail limits of the sequence of measures, and finding necessary and sufficient conditions based on the product measures (hence unverifiable) for the weak convergence of the convolution products for compact groups and certain compact semigroups.

In Chapter 3 verifiable conditions are derived, based only on the individual measures, that are sufficient for the weak convergence of the convolution products in compact abelian and general compact semigroups. It is shown, for example, that if $K$ is the kernel of a compact abelian semigroup $S$, and ($\mu\sb{n}$) is a sequence of probability measures on $S$, then if for each $x$ in $K$ and any open set $N(x)$ containing $x$, $$\sbsp{n\to\infty}{\lim\inf} \mu\sb{n}(N(x)) > 0,$$ then $\mu\sb{k,n} = \mu\sb{k+1} * \mu\sb{k+2} *\cdots* \mu\sb{n}$ converges weakly for all $k \geq 0$ as $n \to \infty.$ The dissertation concludes with the consideration of certain examples showing applications of the theory.