Limit theorems for Markov chains of finite rank (Q799314)

The authors consider a Markov chain \(\{X_ n\}\) on the real line, with a transition distribution function of the form \(P(X_ n\leq y| X_{n- 1}=x)=\sum^{r}_{1}a_ j(x)B_ j(y)\) for real values \(a_ j(x)\) and measures \(B_ j(y)\). Limit theorems for \(X_ n\), for \(X_ 1+...+X_ n\) and for \(\max (X_ 1,...,X_ n)\) are derived. The simplicity of analysis of this model, using matrix theory and the spectral decomposition, is stressed. This relies on such results as for example, that \(P(X_ n\in E| X_ 0=x)=a^ T(x)C^{n-1}B(E)\) where \(a^ T=(a_ 1,...,a_ n)\), \(B^ T=(B_ 1,...,B_ n)\), and C is the matrix with elements \(C_{jk}=\int a_ k(x)B_ j(dx)\).