Geodesic
Random walks on the semi-axis. II.~Limit distributions of boundary functionals
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 3, pp. 464-479

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We prove some limit theorems for the joint distributions of values $\tau_z,x_{\tau_z},i_{\tau_z}(z\to\infty)$, where $\tau_z=\inf\{t\colon x_t\ge z\}$ and $(i_t,x_t)$, $t\ge 0$, is the homogeneous Markov–Feller process in the phase space $\{1,\dots,d\}\times[0,\infty)$ which is additive in the second component and has no negative jumps. 
@article{TVP_1981_26_3_a1,
     author = {V. M. \v{S}urenkov},
     title = {Random walks on the semi-axis. {II.~Limit} distributions of boundary functionals},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {464--479},
     publisher = {mathdoc},
     volume = {26},
     number = {3},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_3_a1/}
}
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V. M. Šurenkov. Random walks on the semi-axis. II.~Limit distributions of boundary functionals. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 3, pp. 464-479. http://geodesic.mathdoc.fr/item/TVP_1981_26_3_a1/