A general principle for limit theorems in finitely additive probability

Karandikar, Rajeeva L.

doi:10.1090/S0002-9947-1982-0667159-6

A general principle for limit theorems in finitely additive probability
HTML articles powered by AMS MathViewer

by Rajeeva L. Karandikar

Trans. Amer. Math. Soc. 273 (1982), 541-550

DOI: https://doi.org/10.1090/S0002-9947-1982-0667159-6

PDF | Request permission

Abstract:

In this paper we formulate and prove a general principle which enables us to deduce limit theorems for sequences of independent random variables in a finitely additive setting from their analogues in the conventional countably additive setting.

References

D. J. Aldous, Limit theorems for subsequences of arbitrarily-dependent sequences of random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), no. 1, 59–82. MR 455090, DOI 10.1007/BF00535707
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
Robert Chen, A finitely additive version of Kolmogorov’s law of the iterated logarithm, Israel J. Math. 23 (1976), no. 3-4, 209–220. MR 407947, DOI 10.1007/BF02761801
Robert Chen, On almost sure convergence in a finitely additive setting, Z. Wahrsch. Verw. Gebiete 37 (1976/77), no. 4, 341–356. MR 571674, DOI 10.1007/BF00533425
Lester E. Dubins, On Lebesgue-like extensions of finitely additive measures, Ann. Probability 2 (1974), 456–463. MR 357724, DOI 10.1214/aop/1176996660
Lester E. Dubins and Leonard J. Savage, How to gamble if you must. Inequalities for stochastic processes, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0236983
Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
Jacques Neveu, Mathematical foundations of the calculus of probability, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. Translated by Amiel Feinstein. MR 0198505
Roger A. Purves and William D. Sudderth, Some finitely additive probability, Ann. Probability 4 (1976), no. 2, 259–276. MR 402888, DOI 10.1214/aop/1176996133
S. Ramakrishnan, Central limit theorems in a finitely additive setting, Illinois J. Math. 28 (1984), no. 1, 139–161. MR 730717
V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 175194, DOI 10.1007/BF00534910