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Short Communications

Conditions for convergence of the superposition of stochastic processes in J-topology

D. S. Sil'vestrov

Kiev

Abstract: 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.

Received: 20.10.1971

 English version: Theory of Probability and its Applications, 1974, 18:3, 579–582

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