Remarks on summability of series formed of deviation probabilities of sums of independent identically distributed random variables

We make some remarks leading to a refinement of the recent work of Klesov (1993) on the connection between the convergence of the series $Sigma _{n = 1}^infty tau _n P(|S_n | ge varepsilon n^alpha )$ for every ε > 0 and the convergence of $Sigma _{n = 1}^infty ntau _n P(|X_1 | ge varepsilon n^alpha )$ again for every ε > 0.