Geodesic
On property of ``waiting process''
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 1, pp. 203-206

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Let $X_1,X_2,\dots$ be i.i.d. integer random variables with $-\infty\le\mathbf EX_10$ and $\mathbf P\{X_1>0\}>0$. Consider the process $W_t$, $t=0,1,\dots$, defined by formula: $$ W_0=0,\quad W_{t+1}=\max\{W_t+X_{t+1};0\}, $$ and its passage time $\tau(N)=\min\{t\colon W_t\ge N\}$, $N=1,2,\dots$. In this paper the existence of $\lim\sqrt[N]{\mathbf E\tau(N)}$ is proved, and its value is found. 
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     author = {V. A. Labkovskii},
     title = {On property of ``waiting process''},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {203--206},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_1_a20/}
}
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V. A. Labkovskii. On property of ``waiting process''. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 1, pp. 203-206. http://geodesic.mathdoc.fr/item/TVP_1973_18_1_a20/