Convergence of series of scalar- and vector-valued random variables and a subsequence principle in 𝐿₂

Dilworth, S. J.

doi:10.1090/S0002-9947-1987-0879579-X

Convergence of series of scalar- and vector-valued random variables and a subsequence principle in $L_ 2$
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by S. J. Dilworth PDF

Trans. Amer. Math. Soc. 301 (1987), 375-384 Request permission

Abstract:

Let $({d_n})_{n = 1}^\infty$ be a martingale difference sequence in ${L_0}(X)$, where $X$ is a uniformly convex Banach space. We investigate a necessary condition for convergence of the series $\sum {_{n = 1}^\infty {a_n}{d_n}}$. We also prove a related subsequence principle for the convergence of a series of square-integrable scalar random variables.

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