Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] K. R. Parthasarathy, Probability measures on metric spaces, Probability and Math. Statist., no. 3, Academic Press, New York, 1967. MR 37 #2271. MR 0226684 (37:2271)
  • [2] B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, GITTL, Moscow, 1949; English transl., Addison-Wesley, Reading, Mass., 1954. MR 12, 839; MR 16, 52. MR 0062975 (16:52d)
  • [3] Ju. V. Prohorov, Convergence of random processes and limit theorems in probability theory, Teor. Verojatnost. i Primenen. 1 (1956), 177-238=Theory Probability Appl. 1 (1956), 157-214. MR 18, 943. MR 0084896 (18:943b)
  • [4] K. R. Parthasarathy, R. Ranga Rao and S. R. S. Varahan, On the category of indecomposable distributions on topological groups, Trans. Amer. Math. Soc. 102 (1962), 200-217. MR 27 #3010. MR 0153041 (27:3010)
  • [5] D. C. Bossard, Probability on locally compact abelian groups: Sums of independent random variables, Thesis, Dartmouth College, Hanover, N. H., 1967.
  • [6] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [7] U. Grenander, Probabilities on algebraic structures, Wiley, New York; Almqvist & Wiksell, Stockholm, 1963. MR 34 #6810. MR 0259969 (41:4598)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60.08

Retrieve articles in all journals with MSC: 60.08


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

American Mathematical Society