A uniform estimate for the rate of convergence in the multidimensional central limit theorem for homogeneous Markov chains (Q1914764)

A uniform estimate is obtained for the remainder term in the central limit theorem (CLT) for a sequence of random vectors \(f(x_1), f(x_2), \dots\) forming a homogeneous Markov chain with arbitrary set of states. This estimate is obtained for \(\sup_{A\in B^k_0} |P_n- \Phi (A)|\) without assumption of the finiteness of \(\sup_{\xi\in X} \int_X |f(\eta) |^3 P(\xi, d\eta)\). CLT for sequences of random vectors \(f(x_1), f(x_2), \dots\) with condition of finiteness of the absolute second moment of the transition probabilities is proved, too.