## ON SOME PROBLEMS IN THE THEORY OF NON-HOMOGENEOUS MARKOV CHAINS

### Abstract (summary)

Some general and well-known definitions are properties of homogeneous Markov chains are first given. By extending these definitions to non-homogeneous Markov chains some general properties of the latter are derived. Most of the extensions deal with essential, inessential, recurrence and periodic properties of the state space of a non-homogeneous Markov chain. Whenever the non-homogeneous analogue of a well-known homogeneous chain property fails, a counterexample is usually given.

Next, Doeblin's theorem giving the ergodic behaviour of a certain class of non-homogeneous chains is given. Some well-known theorems on weak ergodicity are stated. Then Doeblin's condition is relaxed to give Doeblin-like results for a class of non-homogeneous chains which contain, as a subset, the class of chains satisfying Doeblin's condition. Some examples are discussed to justify the assumptions involved.

The final chapter considers an extension of some properties of the basis of a convergent non-homogeneous Markov chain from a finite state space to a countably infinite state space. Some known examples are discussed to explain the modifications in the latter case.