Estimation of finite probabilities via aggregation of Markov chains

An aggregated (trivial) chain with fewer number of states than for the initial Markov chain is constructed such that the finite probabilities of aggregated states equal the finite probabilities of the corresponding states of the initial Markov chain. A method is developed for determining the upper and lower estimates of finite probabilities of aggregated states from data defining the initial Markov chain. These estimates are related with the necessary and sufficient conditions for the classical aggregation of Markov chains. An example on computations is given.