Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations. III

Conditions are found which are necessary for uniform normal convergence in the zones $[0,n^\alpha\rho(n)],[- n^\alpha\rho(n)]$ for values of $\alpha \in[\frac{1}{6},\frac{1}{2}]$, and sufficientin the zones $[0,{n^\alpha}/{\rho}(n)],[{-n^\alpha}/{\rho}(n),0]$. The method is a combination of H. Cramer’s method and of the arguments of Part I of this paper.