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

D. S. Sil'vestrov

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 605-608

Résumé

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.