I. S. Borisov

Exponential inequalities for the distributions of canonical U- and V-statistics of dependent observations

Exponential inequalities are obtained for the distribution tails of canonical (degenerate) U- and V-statistics of an arbitrary order based on samples from a stationary sequence of observations satisfying ϕ-mixing.

Constructing a Stochastic Integral of a Nonrandom Function without Orthogonality of the Noise

In this paper the construction of a stochastic integral of a nonrandom function is suggested without the classical orthogonality condition of the noise. This construction includes some known constructions of univariate and multiple stochastic integrals. Conditions providing the existence of this integral are specified for noises generated by random processes with nonorthogonal increments from certain classes which are rich enough.

A note on Poisson approximation of rescaled set-indexed empirical processes☆

The total variation distance is estimated between distributions of the so-called rescaled empirical process and a Poisson point process which are indexed by the all Borel subsets of a bounded Borel set in

A Note on the Distribution of the Number of Crossings of a Strip by a Random Walk

This paper is essentially an improved result from [V. I. Lotov and N. G. Orlova, Sb. Math., 194 (2003), pp. 927–939], where a formula was obtained for the distribution of the number of crossings of a strip by paths of a random walk defined by an infinite sequence of the partial sums of independent random variables having a common “two-sided geometric” distribution.

Stochastic integrals and asymptotic analysis of canonical von Mises statistics based on dependent observations

In the first part of the paper we study stochastic integrals of a nonrandom function with respect to a nonorthogonal Hilbert noise defined on a semiring of subsets of an arbitrary nonempty set. In the second part we apply this construction to study limit behavior of canonical (i.e., degenerate) Von Mises statistics based on weakly dependent stationary observations.

Probability Inequalities and Limit Theorems for Generalized L-Statistics

We obtain exponential upper bounds for tails of distributions of generalized L-statistics based on a sample from an exponential distribution. We prove the asymptotic normality of generalized L-statistics based on a sample from the uniform distribution on [0,1] and of L-statistics with decomposed kernels (without any restrictions on the sample distribution type).

MOMENT INEQUALITIES CONNECTED WITH ACCOMPANYING POISSON LAWS IN ABELIAN GROUPS

We obtain exact inequalities which connect moments of some functions of sums of independent random variables taking values in a measurable Abelian group and those for the accompanying infinitely divisible laws. Some applications to empirical processes are studied.

Constructing multiple stochastic integrals on non-Gaussian product measures

We study a construction of multiple stochastic integrals of nonrandom functions with respect to the product measures generated by stochastic processes admitting representations as multiple orthogonal random series. This construction is compared with some classical schemes of constructing stochastic integrals of such a kind.

THE FUNCTIONAL LIMIT THEOREM FOR THE CANONICAL U-PROCESSES DEFINED ON DEPENDENT TRIALS

The functional limit theorem is proven for a sequence of normalized U-statistics (the socalled U-processes) of arbitrary order with canonical (degenerate) kernels defined on samples of φ-mixing observations of growing size. The corresponding limit distribution is described as that of a polynomial of a sequence of dependent Wiener processes with some known covariance function.

A Note on Dobrushin's Theorem and Couplings in Poisson Approximation in Abelian Groups

A more general version of Dobrushins result connected with an optimal coupling of two random variables is proven. An application to the problem of Poisson approximation in Abelian groups is considered. In particular, an optimal coupling in Poisson approximation of empirical processes is studied.