Geodesic
Conditions for convergence of the superposition of stochastic processes in J-topology
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 605-608

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Let $\zeta_\varepsilon(t)$, $t\ge0$, and $\nu_\varepsilon(t)$, $t\in[0,T]$, Ьe right-continuous stochastic processes without discontinuities of the second kind. The paper investigates conditions of convergence in J-topology of the superposition of these processes, $\zeta_\varepsilon(\nu_\varepsilon(t))$, $t\in[0,T]$. In the case $\nu_\varepsilon(t)=t$, $t\in[0,T]$, with probability 1 these conditions coincide with well-known Skorohod's conditions of convergence of stochastic processes in J-topology. The results obtained are applied to processes of stepped sums of a random number of random variables. 
@article{TVP_1973_18_3_a16,
     author = {D. S. Sil'vestrov},
     title = {Conditions for convergence of the superposition of stochastic processes in {J-topology}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {605--608},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a16/}
}
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D. S. Sil'vestrov. Conditions for convergence of the superposition of stochastic processes in J-topology. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 605-608. http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a16/