Integrability of infinite sums of independent vector-valued random variables

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 ${S_n} = \Sigma _{j = 1}^n{X_j},M = {\sup _n}|{S_n}|$ 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 $\left \| {{X_n}} \right \|$ such that $E(\varphi (M))$ or $E(\varphi (\left \| S \right \|))$ 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 $P\{ {\varepsilon _1} = 1\} = P\{ {\varepsilon _1} = - 1\} = 1/2$. 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.