Geodesic
On the moment of crossing the one-sided nonlinear boundary by sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 643-656 
A. A. Novikov. On the moment of crossing the one-sided nonlinear boundary by sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 643-656. http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a1/
@article{TVP_1982_27_4_a1,
     author = {A. A. Novikov},
     title = {On the moment of crossing the one-sided nonlinear boundary by sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {643--656},
     year = {1982},
     volume = {27},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the distribution of the stopping time $\tau=\inf\{n\ge 1:S_n\ge f(n)\}$, where $S_n$ is a sum of $n$ independent identically distributed random variables with zero mean. It is shown that for nondecreasing boundaries $f(n)$ the asymptotics of $\mathbf P\{\tau>n\}$ as $n\to\infty$ coinsides (up to some constant) with the asymptotics of the same probability corresponding to $f(n)=\mathrm{const}$ if $\mathbf E(S_1^+)^2<\infty$. An analogous result is obtained in the case of nonincreasing boundaries under the assumption $\mathbf E\operatorname{exp}(\lambda S_1)<\infty$ for some $\lambda>0$. We obtain also the asymptotics of $\mathbf P\{\tau>n\}$ in the case when $\mathbf E\tau<\infty$ and the probability $\mathbf P\{S_1<-x\}$ is regularly varying as $x\to\infty$.