Geodesic
Conditioned stable random walk with a negative drift
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 1, pp. 191-198 
V. I. Afanas'ev. Conditioned stable random walk with a negative drift. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 1, pp. 191-198. http://geodesic.mathdoc.fr/item/TVP_1979_24_1_a19/
@article{TVP_1979_24_1_a19,
     author = {V. I. Afanas'ev},
     title = {Conditioned stable random walk with a~negative drift},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {191--198},
     year = {1979},
     volume = {24},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_1_a19/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $(S_n, n\ge 0)$ be a random walk with a negative drift, $T=\min\{n\colon S_n\le 0\}$. We prove that if the Cramer's type conditions are satisfied then there exists a constant $\Delta>0$ such that the random functions $S_{[nt]}/ \Delta n^{1/2}$, $0\le t\le 1$ considered under the condition $T>n$, converge weakly to a Brownian excursion when $n\to\infty$.