The central limit theorem in a series scheme for a class of interchangeable random variables (Q1912444)

Let \(P\) be a transition probability (stochastic kernel) on a measurable space \((X,B)\), satisfying the conditions of \textit{J. L. Doob} [``Stochastic processes'' (1953; Zbl 0053.26802)] in order that \(P\) has a unique stationary distribution \(\pi\). Consider, on \(X^n\), the transition probability taking \((x_1,\dots,x_n)\) into the product of \(n\) factors \(n^{-1}(\sum^n_{j=1}P(x_j,\;))\) and its stationary distribution \(\pi_n\). Let \(\varphi\in L^2(\pi)\) and \((\xi_i)_{i=1,\dots,n}\) be \(\pi_n\) distributed. Then the distribution of \(n^{-1/2}\sum^n_{i=1}(\varphi(\xi_i)-E\varphi(\xi_i))\) tends to a Gaussian one with null mean and a dispersion equal to \(\sum^\infty_{j=0}|H^j\varphi|^2-\pi_2(\varphi)\), where \(H^j(x,\;)=P^j(x,\;)-\pi\). This is the main theorem, which is proved using characteristic functions.