The limit law of the iterated logarithm in Banach space

In a recent paper by Chen (in press) the limit law of the iterated logarithm for the partial sums of i.i.d. real-valued random variables has been established. In this note we look at the corresponding problem in Banach space setting. Let (B,‖⋅‖) be a real separable Banach space with topological dual B∗. Let {X,Xn;n≥1}{X,Xn;n≥1} be a sequence of i.i.d. B-valued random variables and set Sn=∑i=1nXi,n≥1 and Lt=log(t∨e),LLt=L(Lt),t≥0Lt=log(t∨e),LLt=L(Lt),t≥0. We show thatlimn→∞12LLnmax1≤k≤n‖Sk‖k=σ(X)a.s. provided that lim supn→∞‖Sn‖2nLLn=σ(X)<∞a.s. , where σ2(X)=supf∈B1∗Ef2(X) and B1∗ is the unit ball of B∗.