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Transactions of the American Mathematical Society

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



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: Let B be a normed vector space (possibly a Banach space, but it could be more general) and $ \{ {X_n}\} $ a sequence of B-valued independent random variables on some probability space. Let $ {S_n} = \Sigma _{j = 1}^n{X_j},M = {\sup _n}\vert{S_n}\vert$ and $ S = {\lim _n}{S_n}$ is norm, whenever it exists. Assuming that S exists or $ M < \infty $ a.s. and given certain nondecreasing functions $ \varphi $, we find conditions in terms of the distributions of $ \left\Vert {{X_n}} \right\Vert$ such that $ E(\varphi (M))$ or $ E(\varphi (\left\Vert S \right\Vert))$ is finite.

Let $ \{ {u_n}\} $ be a sequence of elements in B and $ \{ {\varepsilon _n}\} $ a sequence of independent, identically distributed random variables such that $ P\{ {\varepsilon _1} = 1\} = P\{ {\varepsilon _1} = - 1\} = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. We prove some comparison theorems which generalize the following contraction principle of Kahane: If $ \{ {\lambda _n}\} $ is a bounded sequence of scalars, then $ \Sigma {\varepsilon _n}{u_n}$ converges in norm a.s. (or is bounded a.s.) implies the corresponding conclusion for the series $ \Sigma {\lambda _n}{\varepsilon _n}{u_n}$. 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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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