Convergence of weighted sums of random functions in D[0, 1]

Conditions are investigated which imply the tightness of certain weighted sums [Sigma]i = 1kn aniXi of random functions (Xn) taking values in D([0, 1]; E), where E is a separable Banach space. Improved weak laws of large numbers result as corollaries. Examples are presented to clarify the relative strengths of the moment conditions and their relationship to tightness and the strong law of large numbers. A tightness condition is defined using a certain class of sets measurable in the Skorokhod J1-topology, which yields J1-tightness of sequences of weighted sums. As a consequence, tightness of a sequence (Xn) in the Skorokhod M1-topology is used to obtain J1-tightness of a sequence () of averages and a strong law of large numbers in D(R+).