# V. V. Buldygin's research while affiliated with National Technical University of Ukraine Kyiv Polytechnic Institute and other places

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## Publications (76)

We study strong laws of large numbers for multivariate martingales normalized by linear operators in a finite-dimensional Euclidean space. Corollaries of the general results are considered for martingales under moment restrictions.

The problems of almost sure convergence to zero and almost sure boundedness of operator-normed sums of dierent sequences of random vec- tors are studied. The sequences of independent random vectors, orthogonal random vectors and the sequences of vector-valued martingale-diere nces are considered. General results are applied to the problem of asympt...

We investigate necessary and sufficient conditions for the almost-sure boundedness of normalized solutions of linear stochastic differential equations in R
dand their almost-sure convergence to zero. We establish an analog of the bounded law of iterated logarithm.

We investigate properties of the empirical correlation matrix of a centered stationary Gaussian vector field in various function
spaces. We prove that, under the condition of integrability of the square of the spectral density of the field, the normalization
effect takes place for a correlogram and integral functional of it.

We establish the strong law of large numbers with operator normalizations for vector martingales and sums of orthogonal random vectors. We describe its applications to the investigation of the strong consistency of least-squares estimators in a linear regression and the asymptotic behavior of multidimensional autoregression processes.

Preface. 1. Convex sets in vector spaces. 2. Brunn-Minkowski inequality. 3. Convex polyhedra. 4. Two classical isoperimetric problems. 5. Some infinite-dimensional vector spaces. 6. Probability measures and random elements. 7. Convergence of random elements. 8. The structure of supports of Borel measures. 9. Quasi-invariant probability measures. 10...

This preliminary section of our book is devoted to some general facts and theorems concerning convex sets in vector spaces. They will be helpful in our further considerations. Most theorems presented in the section will be given without proofs. For more information around this topic, we refer the reader to the fundamental monograph by Bourbaki [29]...

One of the central topics in our book can be characterized as follows: we discuss appropriate analogues of the Anderson inequality (see Section 2) and demonstrate applications of these analogues in probability theory, mathematical statistics and infinite-dimensional analysis. In other words, we wish to consider several problems and questions which...

We have already mentioned in the Preface that we are not going to discuss in detail various aspects of the theory of isoperimetric problems. This topic is very wide and is under extensive investigations. The reader who is interested in these questions, can be referred to many books and monographs devoted to the subject, e.g., [18], [47], [72], [85]...

As known, Minkowski was one of the founders of modern theory of convex sets. He established many deep and important facts concerning such sets in a finite-dimensional Euclidean space. In particular. several beautiful results are due to him in connection with properties of convex polyhedra.

This section is devoted to topological and geometrical aspects of the theory of σ-finite Borel measures given on metric spaces. Namely, here we investigate the structure of supports of such measures. Since every nonzero σ-finite measure is equivalent to a probability measure (defined on the same σ-algebra), we may restrict our further consideration...

There are various kinds of convergence of random elements, important from the probabilistic viewpoint. Here we recall some of them. The main attention will be paid to the following types of convergence of random elements:
1)
convergence almost surely (or convergence with probability one);
2)
convergence in probability;
3)
convergence in distributio...

This section is basic for our further considerations and is devoted to those convex sets which lie in finite-dimensional topological vector spaces. As mentioned in the previous section, if E is an arbitrary finite-dimensional (Hausdorff) topological vector space, then E is isomorphic to some Euclidean space R
n
. So we can assume, without loss of g...

The next section of our book will be devoted to various kinds of Borel probability measures given on infinite-dimensional topological vector spaces (in particular, on infinite-dimensional Banach spaces). Among them some widely known classical spaces consisting of sequences of real numbers can be met rather frequently. Therefore, it is reasonable to...

Let F be a topological vector space and let µ be a probability measure on (F, B(F)).

In this section. we investigate the so-called prognosis problem which can be formulated as follows. Suppose that two stochastic processes X and Y are given, such that X is not accessible for a direct observation, but Y is completely accessible. Suppose also that there are some close relationships between these two processes. Then it is natural to e...

Let E be an arbitrary set. Any bijection is usually called a transformation of E. Evidently, all transformations of E constitute a group with respect to the composition operation. This group is sometimes denoted by the symbol Sym(E). Any subgroup of Sym(E) is called a group of transformations of E.

Here we demonstrate how the study of Gaussian stochastic processes is possible by comparing their covariance functions. For this purpose, we need some preordering relations in the class Φ+(T) of real-valued positive functions given on a set T of parameters. As a rule, T will be a separable metric space. In particular, the following situations are o...

It is well known that random motion of various objects can be regarded as a mathematical model of certain physical processes and is investigated by the techniques of modern probability theory. The trajectories of such processes may be rather complicated. A characteristic feature of these trajectories is their significant oscillation. In particular,...

We establish sufficient conditions under which shot-noise fields with a response function of a certain form possess the Levy-Baxter
property on an increasing parametric set.

We establish sufficient conditions for singularity of distributions of shot-noise fields with response functions of a certain
form.

We consider shot-noise fields generated by countably additive stochastically continuous homogeneous random measures with independent
values on disjoint sets. We establish necessary and sufficient conditions under which the shot-noise fields possess the Levy-Baxter
property on fixed and increasing parametric sets.

We establish conditions of the weak convergence of the empirical correlogram of a stationary Gaussian process to some Gaussian process in the space of continuous functions. We prove that such a convergence holds for a broad class of stationary Gaussian processes with square integrable spectral density.

We study properties of an empirical correlogram of a centered stationary Gaussian process. We prove that if the spectral density of the process is square integrable, then there is a normalization effect for the correlogram and integral functionals of it.

Exponential estimates of the tails of supremum distributions are obtained for a certain class of pre-Gaussian random processes. The results obtained are applied to the quadratic forms of Gaussian processes and to processes representable as stochastic integrals of processes with independent increments.

A survey of the works of M. I. Yadrenko in the theory of random fields on finite-dimensional and infinite-dimensional spaces is presented.

This paper is the second part of [ibid. 43, No. 4, 444-451 (1991) resp. ibid. 43, No. 4, 482-489 (1991; Zbl 0744.60038)]. Using the comparison theorems which were proved in the first part, the asymptotic normality of the estimator — in a model of a series of several samples — of the correlation function of a stationary Gaussian random process in sp...

This paper is an extension of [11]. Starting from the results of our first paper we prove by inclusion theorems that bounds for the correlation function of a stationary Gaussian process in the space of continuous functions with weight are strongly consistent and asymptotically normal. We construct the simplest functional confidence intervals in the...

A bound is constructed for the correlation function of a uniform Gaussian random field in the scheme of series with respect to the many samples. Exact properties are established for the bound. It is proved that it is strongly consistent and asymptotically normal in the Hilbert space of functions which are square integrable on Rm with same weight fu...

Conditions for Gaussian sequences to converge to zero with unit probability are examined. A comparison theorem is proved, on the basis of which sufficient conditions are derived for the convergence to zero of Gaussian sequences, including, in particular, the previously known ones.

We give the conditions which ensure the compactness of the probability measuresμ
n, n≥1, generated by Gaussian processes the realizations of which are continuous with unit probability in [0, 1]. We also give the conditions for the uniform convergence of stochastic series of the form Σ
k=1∞
2ξk(t), where the ξk(t) are independent Gaussian processes...

Necessary and sufficient conditions are described for convergence with probability 1 of linear methods for stochastic approximation in a separable Banach space to the root of the corresponding operator equation.

## Citations

... The statements below are the analogs of some facts considered in [3] and [11] . Therefore , we give them without proofs. ...

... For a family H ⊆ bb ∞ (E) and X ∈ H, the game G H [X] is the variant of G[X] in which I is restricted to playing elements of H ↾ X. A variant of Theorem 4.5, Theorem 3.11.5 in [30], can be used to obtain the following: Were H(A) to be +-strategic, that is, whenever α is a strategy for II in G H(A) [X], for some X ∈ H(A), then there is an outcome of α in H(A), then the conclusion of the above theorem would yield the desired contradiction. However, by Theorem 3.11.9 of [30], this is equivalent to H(A) being full. ...

Reference: Madness in vector spaces

... Upper bound: Using the Anderson inequality which holds for symmetric processes (cf. [LRZ95], [BK86]), and taking into account (3), we obtain ...

... After year 2000, there were more publications in this subject, and many researchers focused on this subject. Valery et al published their writing for geometric aspects in this area [7]. Liptser summarized essentials in this subject [8]. ...

... The physical formulation of problems of mathematical physics with random factors was studied by Kampe de Feriet [1]. In the works [2] and [3], a new approach studying the solutions of partial differential equations with random initial conditions was proposed. The authors investigate the convergence in probability of the sequence of function spaces of partial sums approximating the solution of a problem. ...

... This space will be denoted by Prg(Ω). Pre–Gaussian random variables are introduced in [2] ...

... In this paper, we consider a class of centered Gaussian Markov sequences (ξ k ) or, in other words, sequences of random variables defined by the system of recurrence relations (1) ξ 0 = 0, ξ k = α k ξ k−1 + β k γ k , k≥ 1, where (α k ) is a sequence of real numbers, (β k ) a sequence of nonnegative numbers, and (γ k ) a standard Gaussian sequence, that is, a sequence of independent Gaussian random variables with zero mean and unit variance. The aim of this paper is to find conditions for the almost sure convergence of the random series (2) ∞ k=1 ξ k for a centered Gaussian Markov sequence (ξ k , k ≥ 1) defined by (1). Note that the asymptotic properties of realizations of Gaussian Markov sequences are studied in the paper [1]. ...

... These processes provide a natural generalization of Gaussian and sub-Gaussian ones and possess well described properties, which is important for applications. Theory developed for these processes allows to derive many useful bounds for the distribution of various functionals of these processes (see, [5]). The present paper is close to the papers [4,13,14,8] where higher order dispersive equations and the heat equation were studied under ϕ-sub-Gaussian initial conditions. ...

... Let ν(p) = ρ(p, 0); then the space GZ[Q](ν) coincides with the Grand Lebesgue Space Gν. This case was investigated in many works ( [5,8,18,19]). ...

... Note that similar problems are treated in the following monographs and papers: [13], [14], [15], [22], [34], [50], [53]- [55], [61], [63], [66]- [75], [80], [81], [88] [94], [95], [97], [98], [114], [117], [118], [121], [122], [126], [128], [132], [133], [134]. ...