Markov Reccurent Sequences in a Random Environment.

Markov Chains in Random Environment (MCRE) is a well-known type of stochastic processes. We consider a subclass of MCRE — Markov Reccurent Sequences in a Random Environment (MRSRE). MRSRE is a MCRE, satisfying an inhomogenuous reccurent equation $Y(n) = A(n) Y(n-1)+B(n)$.

We assume that $A(n)$ are i.i.d. measureable with respect to the environment random variables. The coefficients $B(n)$, in general case, are dependent and have different distributions. However, we suppose that $B(n)$ are independent of the future and satisfy some conditional moment conditions. More explicit we bound the conditional expectation of $h(B(n))$ with respect to $Y(n-1)$ and the environment. We also suppose that 0 is a special state, possible absorbing.

Natural examples of MRSRE are: branching process in a random environment (BPRE), branching process with immigration in a random environment (BPIRE), bisexual branching process in a random environment (BBPRE), bisexual branching process with immigration in a random environment (BBIPRE), special kinds of maximal branching processes (MBP) and maximal branching processes in a random environment (MBPRE).

We show that a wide spectrum of known for BPRE results (and even some unknown) can be generalized to the case of MRSRE. We introduce the associated random walk $S(n)$.

Then we provide

• the convergence of $Y(n) \operatorname{exp}(-S(n))$ to a non-degenerate random variable a.s. and in $L1$;
• central limit theorem analogues for log $Y(n)$ (with convergence to stable laws);
• functional limit theorem for the corresponding process, conditioned on survival $Y(n)>0$;
• asymptotic of survival probability in the critical case ($S(1)$ has zero mean);
• upper large devation results;
• lower large deviation results;
• functional limit theorem for the trajectories of the process conditioned over large devation event. Moreover, the assumptions for MRSRE are close to that for BPRE. However, for BPIRE, BBPRE, BBPIRE, MBP, MBPRE many those results are new.