Geodesic
Limit distribution for a~random walk with absorption
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 588-592

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Let $\xi_1,\xi_2,\dots$ are independent identically distributed random variables, $\mathbf M\xi_1=0$, $\mathbf D\xi_1=1$ and $$ S_n=n^{-1/2}(\xi_1+\dots+\xi_n),\qquad\nu=\min\{n:S_n0\}. $$ We show that $$ \mathbf P\{S_\nu\mid\nu>n\}\to V(x),\qquad\mathbf P\{S_n\mid\nu>n\}\to 1-e^{-x^2/2}. $$ 
@article{TVP_1980_25_3_a11,
     author = {A. V. Pe\v{c}inkin},
     title = {Limit distribution for a~random walk with absorption},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {588--592},
     publisher = {mathdoc},
     volume = {25},
     number = {3},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a11/}
}
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A. V. Pečinkin. Limit distribution for a~random walk with absorption. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 588-592. http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a11/