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


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.)
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