Probability distributions on locally compact abelian groups

For probability distributions on the real line there are three main theorems on which the entire study of limit theorems for sums of independent random variables is based. These are (1) the Lvy-Khinchin representation of an infinitely divisible distribution, (2) the criteria for weak convergence of such distributions, and (3) Khinchin's theorem on sums of infinitesimal summands stating that these converge weakly if and only if certain associated infinitely divisible laws converge. For a precise statement of these results we refer to Kolmogorov and Gnedenko

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