# zbMATH — the first resource for mathematics

Limit theorems for Markov processes in a rapidly varying random medium. (Russian) Zbl 0539.60074
A two-dimensional random process $$(\zeta(t))_{t\geq 0}=(\xi(t),\eta(t))$$ is considered where $$\xi$$ (t)$$\in Z$$, $$\eta$$ (t)$$\in \{1,...,N\} (N<\infty)$$. This process is assumed to possess the homogeneous Markov property with respect to the first exit time from an initial state of the second component ($$\Delta)$$. Let $$(P_{ki})_{k,i\in Z}$$ be a consistent family of probability measures of the process $$\zeta$$ where $$P_{ki}(\xi(0)=k$$, $$\eta(0)=i)=1$$. Such a process can be defined by its probabilities of the form $P_{ki}(\xi(t_ 1)=a_ 1,...,\xi(t_ n)=a_ n,\quad \Delta>t_ n,\quad \xi(\Delta)=\ell,\quad \eta(\Delta)=j)$ where $$0<t_ 1<...<t_ n$$; $$\ell,j\in Z$$. It is assumed that $P_{ki}(\Delta \in dt,\quad \eta(\Delta)=j| \xi)=P_{ki}(\Delta \in dt| \xi)P_{ki}(\eta(\Delta)=j| \xi)$
$P_{ki}(\Delta>t| \xi)=\exp(-\epsilon \int^{t}_{0}\lambda_ i(\xi(s))ds),\quad P_{ki}(\eta(\Delta)=j| \xi)=\lambda_{ij}(\ell)/\lambda_ i(\ell)$ when $$\xi(\Delta)=\ell$$ and the component $$\xi$$ is ergodic. The limit behaviour of the random value $$\mu_ 0(t,N,\epsilon)$$ as t,$$N\to \infty$$, $$\epsilon\to 0$$, is investigated, where $$\mu_ 0(t,N,\epsilon)$$ is the number of states of the process $$\eta$$ which are not visited in the interval [0,t]. Convergence of this distribution to a distribution of some sum of independent random variables is proved. General results are applied to a branching process in a random environment. They allow to represent it as an ordinary branching process.
Reviewer: B.P.Kharlamov

##### MSC:
 60J27 Continuous-time Markov processes on discrete state spaces 60F05 Central limit and other weak theorems 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
##### Keywords:
homogeneous Markov property; ergodic; random environment
Full Text: