On convergence to the Poisson law for distributions of the sum of random variables connected into a chain (Q1925171)

Let \(\{\xi_{nj}\}^n_{j=0}\), \(n\in\mathbb{N}\), be a triangular array of finite Markov chains in some measurable space with transition kernels \(P_n(x,\bullet)\) and stationary distributions \(Q_n(\bullet)\). The chains are assumed to be \(L^p\)-regular, in the sense that \[ |P^k_n(\bullet,*)- Q_n(*)|_{L^p\to L^p}\leq\gamma_n\rho^k_n,\quad k,n\in\mathbb{N}, \] holds for some \(\gamma_n>0\) and \(0<\rho_n<1\). The author gives sufficient conditions in terms of \(P_n\), \(Q_n\), \(\rho_n\), and \(\gamma_n\) such that for measurable, square-integrable (w.r.t. \(\mathbb{P}(\xi_{n1}\in dx)\)) real functions \(f_n\) one has an \(n\to\infty\), \[ \text{law }\Biggl(\sum^n_{j=0} f_n(\xi_{nj})\Biggr)\to\text{Poisson law of parameter }b. \] The parameter is given by \(b=\lim_{n\to\infty} n\mathbb{E} f_n(\xi_{n1})\). The special case \(p=\infty\) contains, in particular, the classical Poisson convergence criterion for i.i.d. random variables.