Integrability of infinite sums of independent vector-valued random variables

Authors:
Naresh C. Jain and Michael B. Marcus

Journal:
Trans. Amer. Math. Soc. **212** (1975), 1-36

MSC:
Primary 60G15; Secondary 60B05, 60G50

MathSciNet review:
0385995

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *B* be a normed vector space (possibly a Banach space, but it could be more general) and a sequence of *B*-valued 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 Hoffmann-Jørgensen. 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:
https://doi.org/10.1090/S0002-9947-1975-0385995-7

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