Convergence rates in the law of logarithm of random elements

We discuss the convergence rates in the law of logarithm for partial sums and randomly indexed partial sums of independent random variables in Banach space, and find the necessary and sufficient conditions on the convergence rates. The results of [1-3] for sums of i.i.d. real valued r.v.,s are extended; Yang's[4] result is generalized and the necessity part of Yang's result is also discussed; a conjecture for the i.i.d. real-valued r.v.s. of [5] is answered in Banach space.