On the Almost Sure Summability of B-Valued Random Variables

Abstract

Since the origin of Probability in Banach spaces, many papers have been devoted to the following problem: “If \(X=(X_{k})_{k\geq1}\) is a sequence of scalar valued random variables (r.v.), which are independent and identically distributed, then the strong law of large numbers (SLLN) (under various forms: Kolmogorov, Erdös-Hsu-Robbins, Marcinkiewicz-Zygmund,…), the central-limit theorem (CLT) or the law of the iterated logarithm (LIL) hold for X as soon as a suitable integrability condition ℑ(∣X 1∣) is fulfilled. If now the X k take their values in a real separable Banach space (B, ∥ ∥) — equipped with its Borel σ-field B — does the condition ℑ(∥X 1∥) also characterize the SLLN, the CLT or the LIL for X?”KeywordsBanach SpaceIterate LogarithmSmooth Banach SpaceSummability MethodIndependent CopyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.