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

In this paper a uniform estimate is obtained for the remainder term in the central limit theorem (CLT) for a sequence of random vectors forming a homogeneous Markov chain with arbitrary set of states. The result makes it possible to estimate the rate of convergence in the CLT without assuming the finiteness of the absolute third moment of the transition probabilities. Some consequences are also proved.