Department of P r o b a b i l i t y and S t a t i s t i c sThe U n i v e r s i t y , Shef f i e l d 53 7RH, U.K. ABSTRACT P o i n t w i s e Weak Law of Large Numbers and Weak Law of Large Numbers i n t h e norm topology of D[O,l] a r e shown t o be e q u i v a l e n t under uniform convex t i g h t n e s s and uniform i n t e g r a b i l i t y c o n d i t i o n s f o r weighted sums of a sequence of random elements i n D[0,1]. Uniform convex t i g h t n e s s and uniform i n t e g r a b i l i t y c o n d i t i o n s a r e j o i n t l y c h a r a c t e r i z e d . Marcinkiewicz-Zygmund-Kolmogorov's and BrunkChung's Strong Laws of Large Numbers a r e d e r i v e d i n t h e s e t t i n g of D[O,l]-space under uniform convex t i g h t n e s s and uniform i n t e g r ab i l i t y c o n d i t i o n s . Equivalence of p o i n t w i s e convergence, convergence i n t h e Skorokhod topology and convergence i n t h e norm topology f o r sequences i n D[O,l] i s s t u d i e d .
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