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November, 1983 The Class of Limit Laws for Stochastically Compact Normed Sums
William E. Pruitt
Ann. Probab. 11(4): 962-969 (November, 1983). DOI: 10.1214/aop/1176993445


Khintchine showed that every infinitely divisible law can be obtained as the limit of a subsequence of normed sums of independent, identically distributed random variables. Here we restrict the summands to be in a class which makes the normed sums stochastically compact, i.e. so that every subsequence has a further subsequence which converges to a nondegenerate limit. A nice analytic condition for stochastic compactness was obtained by Feller. Our result is an analogous characterization of the class of limit laws of subsequences of stochastically compact normed sums. One consequence is that they have $C^\infty$ densities.


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William E. Pruitt. "The Class of Limit Laws for Stochastically Compact Normed Sums." Ann. Probab. 11 (4) 962 - 969, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0519.60014
MathSciNet: MR714959
Digital Object Identifier: 10.1214/aop/1176993445

Primary: 60F05

Keywords: class $\mathscr{L}$ , Infinitely divisible laws , weak convergence

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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