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 { V m 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 { T n , n ≥ 1 } and { τ n , n ≥ 1 } , the main result provides conditions for a weak law of large numbers of the form ∑ i = 1 T m ∑ j = 1 τ n ( V i j − c ( m , n , i , j ) ) / β ( m , n ) → 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 = 1 m ∑ j = 1 n V i j , m ≥ 1 , n ≥ 1 .