Integrability of infinite sums of independent vectorvalued random variables
Authors:
Naresh C. Jain and Michael B. Marcus
Journal:
Trans. Amer. Math. Soc. 212 (1975), 136
MSC:
Primary 60G15; Secondary 60B05, 60G50
MathSciNet review:
0385995
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Abstract: Let B be a normed vector space (possibly a Banach space, but it could be more general) and a sequence of Bvalued independent random variables on some probability space. Let and is norm, whenever it exists. Assuming that S exists or a.s. and given certain nondecreasing functions , we find conditions in terms of the distributions of such that or is finite. Let be a sequence of elements in B and a sequence of independent, identically distributed random variables such that . We prove some comparison theorems which generalize the following contraction principle of Kahane: If is a bounded sequence of scalars, then converges in norm a.s. (or is bounded a.s.) implies the corresponding conclusion for the series . Some generalizations of this contraction principle have already been carried out by HoffmannJørgensen. All these earlier results are subsumed by ours. Applications of our results are made to Gaussian processes, random Fourier series and other random series of functions.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197503859957
PII:
S 00029947(1975)03859957
Keywords:
Integrability,
infinite series,
Banach space valued random variables,
convergence in norm,
contraction principle,
uniformly nondegenerate,
Rademacher sequence
Article copyright:
© Copyright 1975
American Mathematical Society
