Geodesic
Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations.~III
Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 2, pp. 121-134

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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. 
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     author = {Yu. V. Linnik},
     title = {Limit {Theorems} for {Sums} of {Independent} {Variables} {Taking} into {Account} {Large} {Deviations.~III}},
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Yu. V. Linnik. Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations.~III. Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 2, pp. 121-134. http://geodesic.mathdoc.fr/item/TVP_1962_7_2_a0/