Monday, Apr 3, 2017 - 3:30pm - Allen 14
Strong laws of large numbers for double sums of Banach space valued random elements
Robert Parker, Mathematics, University of Florida
Title: Strong laws of large numbers for double sums of Banach space valued random elements
Abstract: The strong law of large numbers (SLLN) is a fundamental result in probability theory. In this talk, we will discuss a generalization of the SLLN to normed double sums of Banach space valued random elements. This setting naturally occurs in areas such as statistical physics. Here, we will present a general strong law of large numbers. We will show that for a double array of independent mean 0 random elements in a real separable Banach space and for a double array of increasing and unbounded positive constants, that a Kolmogorov type condition implies a very general maximal strong law of large numbers and convergence in mean of order p if and only if the Banach space is of Rademacher type p. We will also discuss a 0 < p < 1 and a Brunk-Chung version of this strong law. Throughout, we will present examples that illustrate the sharpness of this result and compare it with other SLLN in the literature. Finally, we will also show that that our result can be used to greatly strengthen a SLLN for sequences of random elements.