Convergence in distribution for randomly stopped random fields

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