Celebratio Mathematica — Varadhan

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K. R. Parthasarathy, R. Ranga Rao, and S. R. S. Varadhan: “On the category of indecomposable distributions on topological groups,” Trans. Amer. Math. Soc. 102 (1962), pp. 200–217. MR 0153041 Zbl 0104.36205 article

According to a theorem of A. I. Khinchine [1937] any distribution on the real line can be written as the convolution of two distributions one of which is the convolution of a finite or a countable number of indecomposable distributions and the other is an infinitely divisible distribution without indecomposable factors. Further, any distribution which is not infinitely divisible has an indecomposable component. This result gives an indication of the existence of a large collection of indecomposable distributions. It is, however, not clear from this result alone that there exists a non-atomic or absolutely continuous indecomposable distribution. This question was raised by H. Cramér [1948] and an answer in the affirmative was given by P. Lévy [1952]. However, what is available is only a meagre supply of examples [Lévy 1952; Dugué 1951; Dugué and Fisher 1948] even in the case of the real line. In this connection there arises naturally the question of the “size” of the class \( \mathcal{M}_1 \),of indecomposable distributions. Or more precisely, what is the category of \( \mathcal{M}_1 \)? The object of this paper is to answer questions of this type.

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K. R. Parthasarathy	Related

K. R. Parthasarathy, R. Ranga Rao, and S. R. S. Varadhan: “Probability distributions on locally compact abelian groups,” Illinois J. Math. 7 : 2 (1963), pp. 337–369. MR 0190968 Zbl 0129.10902 article

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

the Lévy–Khinchin representation of an infinitely divisible distribution,
the criteria for weak convergence of such distributions, and
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 [Kolmogorov and Gnedenko 1954]. During the last two decades or so these results have been extended by many authors to varying degrees of generality. We mention in particular the works of Lévy [1939], Kawada and Itô [1940], Takano [1956], Bochner [1955; 1958], Hunt [1956], Urbanik [1958; 1960], Kloss [1961].

In this paper we study probability distributions on a locally compact abelian (separable) group and obtain definitive extensions of all the three main results mentioned above.

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K. R. Parthasarathy	Related