Convergence in distribution for randomly stopped random fields

Let $\mathbb {X}$ and $\mathbb {Y}$ be two complete, separable, metric spaces, $\xi _\varepsilon (x), x \in \mathbb {X}$ and $\nu _\varepsilon$ be, for every $\varepsilon \in [0, 1]$, respectively, a random field taking values in space $\mathbb {Y}$ and a random variable taking values in space $\mathbb {X}$. We present general conditions for convergence in distribution for random variables $\xi _\varepsilon (\nu _\varepsilon )$ that is the conditions insuring holding of relation, $\xi _\varepsilon (\nu _\varepsilon ) \stackrel {\mathsf {d}}{\longrightarrow } \xi _0(\nu _0)$ as $\varepsilon \to 0$.