Some Strong and Weak Laws of Large Numbers in D(0,1)

Strong laws of large numbers for a sequence x sub n of random functions in D(0,1) are derived using new pointwise conditions on the first absolute moments, which improve on known results. In particular, convex tightness is not implied by the hypotheses of the theorems. It is shown that convex tightness is is preserved when random functions are centered, and this result is applied to improve some known strong laws for weighted sums in D(0,1). A weak law of large numbers is proved using a new pointwise condition on the first moments and some weak laws for weighted sums are improved upon by weakening the hypotheses. A study is made of relationships among several conditions on X sub n which appear as hypotheses in laws of large numbers. (Author) Prepared in cooperation with Louisiana Tech. Univ., Ruston., Dept. of Mathematics and Statistics.