Vladimir V. Ulyanov

On the accuracy of normal approximation

The aim of the present paper is to obtain estimates of the speed of convergence in the central limit theorem in Rk for variation distance valid when (truncated) pseudo-moments are small enough. Together with the integral type estimates of Bhattacharya and Sweeting [5,6] the results of this paper lead to the integral type estimates in terms of pseudo-moments. Similar (but somewhat less general) results were anounced in [1].

Nonuniform error bounds in asymptotic expansions for scale mixtures under mild moment conditions

Let F(x) be a distribution function of of a scale mixture X=SZ of a random variable Z with distribution G and scale factor S, which is a positive random variable independent of Z. Some nonuniform bounds are given for asymptotic expansions of F(x) around G(x)_ under mild moment conditions on the distribution of S. Some nonuniform bounds for the normal approximation to the Student t-distribution are given as examples.

Some Approximation Problems in Statistics and Probability

We review the results about the accuracy of approximations for distributions of functionals of sums of independent random elements with values in a Hilbert space. Mainly we consider recent results for quadratic and almost quadratic forms motivated by asymptotic problems in mathematical statistics. Some of the results are optimal and could not be further improved without additional conditions.

Characterization and Stability Problems for Finite Quadratic Forms

Sufficient conditions are given under which the distribution of a finite quadratic form in independent identically distributed symmetric random variables defines uniquely the underlying distribution. Moreover, a stability theorem for quadratic forms is proved.

Asymptotic Analysis of Symmetric Functions

In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed.

On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data

We study the rate of weak convergence of the distributions of the statistics {tλ(Y), λ ∈ ℝ} from the power divergence family of statistics to the χ2 distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that

Accurate Approximation of Correlation Coefficients by Short Edgeworth-Chebyshev Expansion and Its Statistical Applications

\Pr (t_\lambda (Y) < c) = G_2 (c) + O(n^{ - 50/73} (\log n)^{315/146} ),

On Properties of Polynomials in Random Elements

where G2(c) is the χ2 distribution function of a random variable with two degrees of freedom. In the proof we use Huxley’s theorem of 1993 on approximating the number of integer points in a plane convex set with smooth boundary by the area of the set.

On Distribution of Quadratic Forms in Gaussian Random Variables

In Christoph, Prokhorov and Ulyanov (Theory Probab Appl 40(2):250–260, 1996) we studied properties of high-dimensional Gaussian random vectors. Yuri Vasil’evich Prokhorov initiated these investigations. In the present paper we continue these investigations. Computable error bounds of order O(n − 3) or O(n − 2) for the approximations of sample correlation coefficients and the angle between high-dimensional Gaussian vectors by the standard normal law are obtained. We give some numerical results as well. Moreover, different types of Bartlett corrections are suggested.

Asymptotically precise estimate of the accuracy of Gaussian approximation in Hilbert space

The paper deals with different properties of polynomials in random elements: bounds for characteristics functionals of polynomials, stochastic generalization of the Vinogradov mean value theorem, characterization problem, bounds for probabilities to hit the balls. These results cover the cases where the random elements take values in finite as well as infinite dimensional Hilbert spaces.