Weak laws of large numbers for double sums of independent random elements in Rademacher type p and stable type p Banach spaces

Abstract

For a double array {Vm n, m ≥ 1, n ≥ 1} of independent random elements in a real separable stable type p (1 ≤ p < 2) Banach space X and sequences of random positive integers {Tn, n ≥ 1} and {τn, n ≥ 1}, the main result provides conditions for a weak law of large numbers of the form ∑i = 1Tm ∑j = 1τn (Vi j - c (m, n, i, j)) / β (m, n) over(→, P) 0 as max {m, n} → ∞ to hold where the c (m, n, i, j) are suitable elements in X and the β (m, n) are suitable norming constants. The conditions are shown to completely characterize stable type p (1 ≤ p < 2) Banach spaces. Illustrative examples are provided. Moreover, for a double array of independent random elements in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space, a weak law of large numbers is obtained for the double sums ∑i = 1m ∑j = 1n Vi j, m ≥ 1, n ≥ 1. © 2009 Elsevier Ltd. All rights reserved.