Geodesic
Some results connected with the expectation process
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 1, pp. 174-181 
I. Yu. Faсtorovič. Some results connected with the expectation process. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 1, pp. 174-181. http://geodesic.mathdoc.fr/item/TVP_1982_27_1_a18/
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     author = {I. Yu. Fa{\cyrs}torovi\v{c}},
     title = {Some results connected with the expectation process},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {174--181},
     year = {1982},
     volume = {27},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_1_a18/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper we find the necessary and sufficient conditions for the exponentiality of the asymptotics of the average time which is needed for the integer expectation process $W_t(t=0,1,2,\dots)$ to reach the level $N\to\infty$. It is established also that the well-known sufficient condition for the exponentiality of $$ \mu_N=\mathbf P\{\max_{n\ge 0}S_n=N\} $$ (where $S_n$ is a random walk with negative drift) is a necessary one.