Convergence of Weighted Sums and Laws of Large Numbers in D([0,1]; E)

Abstract

Convergence properties of weighted sums of functions in D([0, 1]; E) ( E a Banach space) are investigated. We show that convergence in the Skorokhod J 1-topology of a sequence ( x n ) in D([0, 1]; E) does not imply convergence of a sequence ( x n ) of averages. Convergence in the J 1-topology of a sequence ( x n ) of averages is shown, under the growth condition ∥ x n ∥ ∞ = o( n), to be equivalent to the convergence of ( x n ) in the uniform topology. Convergence of a sequence ( x n ,) is shown to imply convergence of the sequence ( x n ) of averages in the M 1 and M 2 topologies. The strong law of large numbers in D[0, 1] is considered and an example is constructed to show that different definitions of the strong law of large numbers are nonequivalent.