Geodesic
Remarks about the limit of composite random function
Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 4, pp. 707-715 
D. S. Sil'vestrov. Remarks about the limit of composite random function. Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 4, pp. 707-715. http://geodesic.mathdoc.fr/item/TVP_1972_17_4_a7/
@article{TVP_1972_17_4_a7,
     author = {D. S. Sil'vestrov},
     title = {Remarks about the limit of composite random function},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {707--715},
     year = {1972},
     volume = {17},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1972_17_4_a7/}
}
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Let $\xi_{\varepsilon}(t)$, $t\geq 0$, be a continuous from the right stochastic process without discontinuities of the second kind and $\nu_{\varepsilon}$, for each $\varepsilon\geq 0$, be a non-negative random variable. In the paper, general sufficient conditions are studied for weak convergence of the distribution functions of the random variables $\xi_{\varepsilon}(\nu_{\varepsilon})$ to the distribution function of $\varepsilon_0(\nu_0)$ as $\varepsilon\to 0$.