Geodesic
On Convergence of Sums of Step Stochastic Processes to a Poisson Process
Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 2, pp. 189-194

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Let $X_n (t)=\sum_{r-1}^{k_n}X_{nr}(t)$, where $X_{nr}(t)$ are independent asymptotically negligible stochastic processes with non-negative integer-valued increments. The necessary and sufficient conditions for convergence of the sequence $\{X_n(t)\}$ to a given Poisson process are proved. 
@article{TVP_1963_8_2_a5,
     author = {B. Grigelionis},
     title = {On {Convergence} of {Sums} of {Step} {Stochastic} {Processes} to a {Poisson} {Process}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {189--194},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {1963},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1963_8_2_a5/}
}
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B. Grigelionis. On Convergence of Sums of Step Stochastic Processes to a Poisson Process. Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 2, pp. 189-194. http://geodesic.mathdoc.fr/item/TVP_1963_8_2_a5/