Asymptotic property for some series of probability

Let {X,X _n , n ≥ 1 be a sequence of i.i.d.random variables with zero mean, and set $ S_n = \sum\limits_{k = 1}^n {X_k ,EX^2 } = \sigma ^2 > 0,\lambda (\varepsilon ) = \sum\limits_{n = 1}^\infty {P(|S_n | \geqslant n\varepsilon )} $ . In this paper, we discuss the rate of the approximation of σ ² by ? ² λ(?) under suitable conditions, and improve the corresponding results of Klesov (1994).