Studentized increments of partial sums (Q1326987)

Let \(X, X_ 1, X_ 2,\dots\) be i.i.d. random variables with \(EX=0\). Assuming a necessary and sufficient condition for \(X\) to belong to the domain of attraction of the normal law, i.e. \[ \lim_{x\to\infty} (x^ 2 P(| X|> x)/ E X^ 2 I(| X| \leq x)) =0, \] the authors study almost sure limiting properties of Studentized increments of partial sums \(S_ n= \sum_{i=1}^ n X_ i\).