Geodesic
Integral limit theorems taking into account large deviations when Cram\'er's condition does not hold.~II
Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 2, pp. 203-216

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Let $\xi_1,\dots\xi_n,\dots$ be a sequence of independent equally distributed random variables and $\mathbf M\xi_n=0$. The density function $p(x)$ of $\xi_n$ being assumed to satisfy the condition $$ p(x)\sim e^{-|x|^{1-\varepsilon}},\quad0\varepsilon1,\quad\text{as }|x|\to\infty, $$ the behaviour of the probability $\mathbf P\{\xi_i+\dots+\xi_n>x\}$ is studied when $n$ and $x$ tend to infinity so that $x>\sqrt n$. 
@article{TVP_1969_14_2_a1,
     author = {A. V. Nagaev},
     title = {Integral limit theorems taking into account large deviations when {Cram\'er's} condition does not {hold.~II}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {203--216},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1969_14_2_a1/}
}
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A. V. Nagaev. Integral limit theorems taking into account large deviations when Cram\'er's condition does not hold.~II. Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 2, pp. 203-216. http://geodesic.mathdoc.fr/item/TVP_1969_14_2_a1/