Limit theorems for Markov chains of finite rank

We consider a Markov chain with a general state space, but whose behaviour is governed by finite matrices. After a brief exposition of the basic properties of this chain its convenience as a model is illustrated by three limit theorems. The ergodic theorem, the central limit theorem and an extreme-value theorem are expressed in terms of dominant eigenvalues of finite matrices and proved by simple matrix theory. Key words: Markov chains of finite rank, spectral decomposition of matrices, ergodic theorem, central limit theorem, extreme values.