Geodesic
Integral limit theorems taking into account large deviations when Cramer's condition does not hold. I
Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 1, pp. 51-63 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\xi_1,\dots,\xi_n$ be a sequence of independent equally distributed random variables with $\mathbf M\xi_n=0$. Throughout the paper it is supposed that the density function $p(x)$ of $\xi^n$ has the property $$ p(x)\sim e^{-|x|^{1-\varepsilon}},\quad0<\varepsilon<1,\quad|x|\to\infty. $$ The problem we deal with is to describe the behaviour of the probability $\mathbf P\{\xi_1+\dots+\xi_n>x\}$ when $x$ tends to infinity so that $x>\sqrt n$. 
@article{TVP_1969_14_1_a5,
     author = {A. V. Nagaev},
     title = {Integral limit theorems taking into account large deviations when {Cramer's} condition does not {hold.~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {51--63},
     year = {1969},
     volume = {14},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1969_14_1_a5/}
}
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A. V. Nagaev. Integral limit theorems taking into account large deviations when Cramer's condition does not hold. I. Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 1, pp. 51-63. http://geodesic.mathdoc.fr/item/TVP_1969_14_1_a5/