On continuity of stochastic sequences generated by recurrent procedures

Sequences of random variables $\overline Y=\{Y_n\}$ are considered which are constructed from sequences of random variables $\overline X=\{X_n\}$ by the recurrent formula
$$ Y_{n+1}=F(X_n,Y_n);\quad n=1,2,\dots. $$
(All the random variables take values in some metric spaces.) The problem of continuous dependence of $\overline Y$ on $\overline X$ is posed and solved, continuity being understood in the sense of some or other definition of distance in spaces of finite and infinite collections of random variables.
Along with general results related to the problem posed, the paper describes shortly the method of minimal matrices and some other new notions and facts connected with the problem under consideration.
The paper is closely related to continuity problems in queueing theory.