Estimates of Berry–Esseen type for probabilities of large deviations under violation of Cramér condition

Two-sided estimates for probabilities of large deviations for sums of independent random variables with finite variances are obtained. All asymptotics of the probabilities are described in terms of deviation function $\Lambda(x,y)$ of a sum of truncated random variables. All error terms are explicitly estimated by a modified Lyapunov ratio $L(H(x),y)$.