Remarks on the weak limit of the superposition of asymptotically independent random functions

Let $\xi(t)$, $t\ge 0$, be a continuous from the right stochastic process without discontinuities of the second kind and $\nu_{\varepsilon}$ (for each $\varepsilon\ge 0$) be a non-negative random variable. In this paper we study some general sufficient conditions for the weak convergence of the distribution functions of random variables $\xi_{\varepsilon}(\nu_{\varepsilon})$ to the distribution function of $\xi_0(\nu_0)$ as $\varepsilon\to 0$ for the scheme when the process $\xi_{\varepsilon}(t)$ and the variable $\nu_{\varepsilon}$ are asymptotically (as $\varepsilon\to 0$) independent.