Estimating the stationary distribution of a Markov chain

Let <InlineEquation ID="Equ1"> <EquationSource Format="TEX"><![CDATA[$\{X_j\}^\infty_0$]]></EquationSource> </InlineEquation> be a Markov chain with a unique stationary distribution <InlineEquation ID="Equ2"> <EquationSource Format="TEX"><![CDATA[$\pi$]]></EquationSource> </InlineEquation>. Let h be a bounded measurable function. Write <InlineEquation ID="Equ3"> <EquationSource Format="TEX"><![CDATA[$\lambda_{h}=\int hd\pi$]]></EquationSource> </InlineEquation> and <InlineEquation ID="Equ4"> <EquationSource Format="TEX"><![CDATA[$\hat{\lambda}_{hn}=\frac{1}{(n+1)}\sum^n_0h(X_j)$]] ></EquationSource> </InlineEquation>. This paper explores conditions for the <InlineEquation ID="Equ5"> <EquationSource Format="TEX"><![CDATA[$\sqrt{n}$]]></EquationSource> </InlineEquation> consistency and asymptotic normality of the estimate of <InlineEquation ID="Equ6"> <EquationSource Format="TEX"><![CDATA[$\hat{\lambda}_{hn}$]]></EquationSource> </InlineEquation> of <InlineEquation ID="Equ7"> <EquationSource Format="TEX"><![CDATA[$\lambda_h$]]></EquationSource> </InlineEquation> assuming the existence of a solution to the Poisson equation <InlineEquation ID="Equ8"> <EquationSource Format="TEX"><![CDATA[$h - \lambda_h=g-Pg$]]></EquationSource> </InlineEquation>. Our framework covers the case of nonirreducible Markov chains arising in many growth models in economics. Copyright Springer-Verlag Berlin Heidelberg 2003