ON THE LAW OF THE ITERATED LOGARITHM FOR RAPIDLY INCREASING SUBSEQUENCES

The usual law of the iterated logarithm states that the partial sums S -n of independent and identically distributed random variables can be normalized by the sequence a(n) = root n log log n, such that lim sup( n-->infinity) S-n/a(n) = root 2 a.s.. As has been pointed out by GUT ( 1986) the law fails if one considers the limsup along subsequences whi ch increase faster than exponentially. In particular, for very rapidly increasing subsequences {n(k), k greater than or equal to 1} one has lim sup(k-->infinity) S-nk/a(nk) = 0 a.s.. In these cases the normaliz ing constants a(nk) have to be replaced by root n(k) log k to obtain a non-trivial limiting behaviour: lim sup(k-->infinity) S-nk/root n(k) log k = root 2 a.s.. We will present an intelligible argument for this structural change and apply it to related results.