Extreme values and the law of the iterated logarithm (Q1071365)

If X takes values in a Banach space B and is in the domain of attraction of a Gaussian law on B, then X satisfies the compact law of the iterated logarithm (LIL) with respect to a regular normalizing sequence \(\{\gamma_ n\}\) iff X satisfies a certain integrability condition. The integrability condition is equivalent to the fact that the maximal term of the sample \(\{\| X_ 1\|,\| X_ 2\|,...,\| X_ n\| \}\) does not dominate the partial sums \(\{S_ n\}\), and here we examine the precise influence of these maximal terms and its relation to the compact LIL. In particular, it is shown that if one deletes enough of the maximal terms there is always a compact LIL with non-trivial limit set.