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 $\Sigma _{n = 1}^\infty n\tau _n P(|X_1 | \ge \varepsilon n^\alpha )$ again for every ε > 0.