ИСТИНА

We give a review of recent results on one- and two-terms Chebyshev--Edgeworth type and Cornish--Fisher type expansions in approximation problems for distributions of linear and non-linear statistics and its quantiles. High-dimensional and large-sample approximations with non-asymptotic error bounds are considered. The case of random size samples is also discussed. Since Chebyshev—Edgeworth expansions are based on non-monotonic Chebyshev-Hermite polynomials, we demonstrate the rearrangement technique to make the expansions monotonic. It originated in the work of P.L. Chebyshev. At last, we consider the expansions for a class of sequences of symmetric functions of many variables. It implies a general approach to get the non-asymptotic bounds for accuracy of approximation of nonlinear forms in random elements in terms of Lyapunov type ratios.