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

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 J1-topology of a sequence (xn) in D([0, 1]; E) does not imply convergence of a sequence (xn) of averages. Convergence in the J1-topology of a sequence (xn) of averages is shown, under the growth condition ∥ xn ∥ ∞ = o(n), to be equivalent to the convergence of (xn) in the uniform topology. Convergence of a sequence (xn,) is shown to imply convergence of the sequence (xn) of averages in the M1 and M2 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.