Sums of independent random variables with finite variances – moderate deviations and nonuniform bounds in the CLT

In the paper the new nonuniform bounds in the CLT for sums of independent random variables with finite variances are contained. Some simple and sharp nonasymptotic upper bounds for large deviation probabilities of the sums are derived. The results obtained are applied to refine the known conditions and to find the new ones under which the large deviation theorems in the series scheme are true.