TOWARD A GENERAL LAW OF THE ITERATED LOGARITHM IN BANACH-SPACE

A general bounded law of the iterated logarithm for Banach space value d random variables is established. Our result implies: (a) the bounded LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables i n the domain of attraction of a Gaussian law and (c) new LIL results f or random variables outside the domain of attraction of a Gaussian law in cases where the classical norming sequence {root nLLn} does not wo rk. Basic ingredients of our proof are an infinite-dimensional Fuk-Nag aev type inequality and an infinite-dimensional version of Klass's K-f unction.