Geodesic
Large-Deviation Probabilities for Maxima of Sums of Independent Random Variables with Negative Mean and Subexponential Distribution
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 2, pp. 387-397

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We consider the sums $S_n=\xi_1+\dots+\xi_n$ of independent identically distributed random variables with negative mean value. In the case of subexponential distribution of the summands, the asymptotic behavior is found for the probability of the event that the maximum of sums $\max(S_1,\ldots,S_n)$ exceeds high level $x$. The asymptotics obtained describe this tail probability uniformly with respect to all values of $n$.
Keywords: maxima of sums of random variables, homogeneous Markov chain, large deviation probabilities, subexponential distribution, integrated tail distribution. 
@article{TVP_2001_46_2_a14,
     author = {D. A. Korshunov},
     title = {Large-Deviation {Probabilities} for {Maxima} of {Sums} of {Independent} {Random} {Variables} with {Negative} {Mean} and {Subexponential} {Distribution}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {387--397},
     publisher = {mathdoc},
     volume = {46},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2001_46_2_a14/}
}
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D. A. Korshunov. Large-Deviation Probabilities for Maxima of Sums of Independent Random Variables with Negative Mean and Subexponential Distribution. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 2, pp. 387-397. http://geodesic.mathdoc.fr/item/TVP_2001_46_2_a14/