ON THE LIL FOR SELF-NORMALIZED SUMS OF IID RANDOM-VARIABLES

Let X, X-i, i is an element of N, be i.i.d. random variables and let, for each n is an element of N, S-n = Sigma(i=1)(n) X-i and V-n(2) = Si gma(i=1)(n) X-i(2). It is shown that lim sup(n-->infinity)\S-n\/(V-n r oot log log n) < infinity a.s. whenever the sequence of self-normalize d sums S-n/V-n is stochastically bounded, and that this limsup is a.s. positive if, in addition, Xis in the Feller class. It is also shown t hat, for X in the Feller class, the sequence of self-normalized sums i s stochastically bounded if and only if lim sup(t-->infinity)[t\EXI(\X \ less than or equal to t)\/(EXI)-I-2(\X\ less than or equal to t)] < infinity.