A RANDOM WALK AND ITS LIL IN A BANACH SPACE

Let an : n 1} be a sequence of i.i.d. Banach space valued random variables with E[X„]=0 and Ell X.112<00, and let So=0, Sn= XiF X2+ . . . + Xn, n 1. We prove that if {Sn : n_. 1} satisfies the LIL in B then the sequence {77,, : n .1} satisfies the LIL in C([0, 1], B), where 77n(t)=S[nt]+ (nt—[nt]) X[nt]-14, Ot51 and C([°, 1], B) --={ f : [0, 1] ----. BI f is continuous}. We also use this result to give an alternative to the proof of the LIL of Brownian motion in Banach spaces.