Let {Sn, n[greater-or-equal, slanted]1} denote the partial sum of a sequence (Xn) of i.i.d. random variables with mean 0. The equivalent statements of E(supnSn[alpha]/cn) t8 for certain sequences (cn) and positive constants [alpha] are investigated. We find a useful sufficient condition of...

The regularity of trajectories of continuous parameter process (Xt)t[set membership, variant]R+ in terms of the convergence of sequence E(XTn) for monotone sequences (Tn) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems...