On large deviations in estimation problem with dependent observations

Abstract

The paper is devoted to the stochastic optimization problem with a stationary ergodic random sequence satisfying the hypermixing condition. It is assumed that we have the finite number of observed elements in the sequence, and instead of solving the former problem we investigate the empirical function, find its points of minimum, and study their asymptotic properties. More precisely we consider the probabilities of large deviations of minimizers and the minimal value of the empirical criterion function from the corresponding characteristics of the main problem. The conditions under which the probabilities of the large deviations decrease exponentially are found.