Limit theorems for measures on nonmetrizable locally compact abelian groups

Author:
David C. Bossard

Journal:
Trans. Amer. Math. Soc. **159** (1971), 185-205

MSC:
Primary 60.08

DOI:
https://doi.org/10.1090/S0002-9947-1971-0279845-3

MathSciNet review:
0279845

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Abstract: In a recent book, Parthasarathy provides limit theorems for sums of independent random variables defined on a metrizable locally compact abelian group. These results make heavy use of the metric assumption. This paper consists of a reworking of certain results contained in Parthasarathy to see what can be done without the metric restriction. Among the topics considered are: necessary and sufficient conditions for a limit law to have an idempotent factor; the relationship between limits of compound Poisson laws and limits of sums of independent random variables; and a representation theorem for certain limit laws.

**[1]**K. R. Parthasarathy,*Probability measures on metric spaces*, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR**0226684****[2]**B. V. Gnedenko and A. N. Kolmogorov,*Limit distributions for sums of independent random variables*, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR**0062975****[3]**Yu. V. Prokhorov,*Convergence of random processes and limit theorems in probability theory*, Teor. Veroyatnost. i Primenen.**1**(1956), 177–238 (Russian, with English summary). MR**0084896****[4]**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), 200–217. MR**0153041**, https://doi.org/10.1090/S0002-9947-1962-0153041-7**[5]**D. C. Bossard,*Probability on locally compact abelian groups: Sums of independent random variables*, Thesis, Dartmouth College, Hanover, N. H., 1967.**[6]**Edwin Hewitt and Kenneth A. Ross,*Abstract harmonic analysis. Vol. I*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR**551496****[7]**Ulf Grenander,*Probabilities on algebraic structures*, Second edition, Almqvist & Wiksell, Stockholm; John Wiley & Sons Inc., New York-London, 1968. MR**0259969**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0279845-3

Keywords:
Locally compact groups,
nonmetrizable spaces,
Prokhorov Theorem,
conditional compactness of measures,
probability measures,
limit theorems,
infinitesimal arrays,
triangular arrays,
infinitely divisible distributions,
idempotent measures,
characteristic functions,
representation theorems,
compound Poisson laws

Article copyright:
© Copyright 1971
American Mathematical Society