On the LIL for self-normalized sums of IID random variables (Q1266782)

Let \(X\), \(X_i\), \(i\in N\), be i.i.d. real random variables, and set \(S_n= \sum^n_{i=1} X_i\), \(V^2_n= \sum^n_{i=1} X^2_i\). It is shown that \(\limsup_{n\to\infty}| S_n|/V_n \sqrt{\log\log n}< \infty\) a.s. whenever the sequence of self-normalized sums \(| S_n|/V_n\) is stochastically bounded, and that this lim sup is a.s. positive if, in addition, \(X\) is in the Feller class. A criterion for the sequence of self-normalized sums to be stochastically bounded is given.