Том 219, № 5 (2016)
- Год: 2016
- Статей: 19
- URL: https://journals.rcsi.science/1072-3374/issue/view/14791
Article
To the Memory of M. I. Gordin
Several Words About Mikhail Iosifovich Gordin
M. I. Gordin on Doeblin and on Doeblin–Fortet’s Work (Extracts from Letters)
Equivalence of the Brownian and Energy Representations
Аннотация
We consider two unitary representations of infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by the quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent. Bibliography: 28 titles.
Hyperbolic Ornstein–Uhlenbeck Process
Аннотация
In this paper, we continue the study of the class of hypergeometric diffusions started by the author. A broad subclass of these diffusions consists of hyperbolic Ornstein–Uhlenbeck processes. An explicit formula for the transition density of a hyperbolic Ornstein–Uhlenbeck process is derived. Bibliography: 7 titles.
Criteria of Divergence Almost Everywhere in Ergodic Theory
Аннотация
In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples: the Stein continuity principle, Bourgain’s entropy criteria, and Kakutani–Rokhlin lemma, the most classical device for these questions in ergodic theory. First, we state a L1-version of the continuity principle and give an example of its usefulness by applying it to a famous problem on divergence almost everywhere of Fourier series. Next we particularly focus on entropy criteria in Lp, 2 ≤ p ≤ ∞, and provide detailed proofs. We also study the link between the associated maximal operators and the canonical Gaussian process on L2. We further study the corresponding criterion in Lp, 1 < p < 2, using properties of pstable processes. Finally, we consider Kakutani–Rokhlin’s lemma, one of the most frequently used tools in ergodic theory, by stating and proving a criterion for a.e. divergence of weighted ergodic averages. Bibliography: 38 titles.
On the Classification Problem of Measurable Functions in Several Variables and on Matrix Distributions
Аннотация
We resume results of the first author on classification of measurable functions in several variables, with some minor corrections of purely technical nature. We give a partial solution of the characterization problem for so-called matrix distributions which are metric invariants of measurable functions introduced by the first author. Matrix distributions are considered as (Sℕ × Sℕ)-invariant, ergodic measures on the space of matrices; this fact connects our problem with Aldous’ and Hoover’s theorem. Bibliography: 14 titles.
Discriminant and Root Separation of Integral Polynomials
Аннотация
Consider a random polynomial GQ(x) = ξQ,nxn + ξQ,n − 1xn − 1 + ⋯ + ξQ,0 with independent coefficients that are uniformly distributed on 2Q+1 integer points {−Q, . . .,Q}. Denote by D(GQ) the discriminant of GQ. We show that there exists a constant Cn depending on n only such that for all Q ≥ 2, the distribution of D(GQ) can be approximated as follows: \( \underset{-\infty \le a\le b\le -\infty }{ \sup}\left|\mathrm{P}\left(a\frac{D\left({G}_Q\right)}{Q^{2n-2}}\le b\right)-{\displaystyle \underset{a}{\overset{b}{\int }}{\upvarphi}_n(x)dx}\right|\le \frac{C_n}{ \log Q}, \) where \( \varphi \)n denotes the probability density function of the discriminant of a random polynomial of degree n with independent coefficients that are uniformly distributed on [−1, 1]. Let Δ(GQ) denote the minimal distance between complex roots of GQ. As an application, we show that for any ε > 0 there exists a constant δn > 0 such that Δ(GQ) is stochastically bounded from below/above for all sufficiently large Q in the following sense: \( \mathrm{P}\left({\delta}_n<\varDelta \left({G}_Q\right)<\frac{1}{\delta_n}\right)>1-\varepsilon \). Bibliography: 14 titles.
On the Convex Hull and Winding Number of Self-Similar Processes
Аннотация
It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in Rd, its convex hull V (t) = conv{B(s), s ≤ t} with probability 1 for each t > 0 contains 0 as an interior point. We also know that the winding number of a typical path of a two-dimensional BM is equal to +∞. The aim of this paper is to show that these properties are not specifically “Brownian,” but hold for a much larger class of d-dimensional self-similar processes. This class contains, in particular, d-dimensional fractional Brownian motions and (concerning convex hulls) strictly stable Lévy processes. Bibliography: 10 titles.
Circular Unitary Ensembles: Parametric Models and Their Asymptotic Maximum Likelihood Estimates
Аннотация
Parametrized families of distributions for the circular unitary ensemble in random matrix theory are considered; they are connected to Toeplitz determinants and have many applications in mathematics (for example, to the longest increasing subsequences of random permutations) and physics (for example, to nuclear physics and quantum gravity). We develop a theory for the unknown parameter estimated by an asymptotic maximum likelihood estimator, which, in the limit, behavesas the maximum likelihood estimator if the latter is well defined and the family is sufficiently smooth. They are asymptotically unbiased and normally distributed, where the norming constants are unconventional because of long range dependence.
On a Problem of Estimation of Infinite-Dimensional Parameter
Аннотация
Let X be a random variable taking positive integer values and let P{X = k} = θ(k). We consider the problem of estimation of the parameter θ = (θ(1), θ(2), . . . ) on the base of a sample X1,X2, . . . , Xn, where the observations Xj are independent copies of X. Bibliography: 5 titles.
A Bound for the Maximal Probability in the Littlewood–Offord Problem
Аннотация
In this paper, we study a connection of the Littlewood–Offord problem with estimating the concentration functions of some symmetric, infinitely divisible distributions. It is shown that the values at zero of the concentration functions of weighted sums of i.i.d. random variables may be estimated by the values at zero of the concentration functions of symmetric, infinitely divisible distributions with the Lévy spectral measures which are multiples of the sum of delta-measures at ±weights involved in constructing the weighted sums.
Invariance, Quasi-Invariance, and Unimodularity for Random Graphs
Аннотация
We treat the probabilistic notion of unimodularity for measures on the space of rooted, locally finite, connected graphs in terms of the theory of measured equivalence relations. It turns out that the right framework for this consists in considering quasi-invariant (rather than just invariant) measures with respect to the root moving equivalence relation. We define a natural modular cocycle of this equivalence relation and show that unimodular measures are precisely those quasi-invariant measures whose Radon–Nikodym cocycle coincides with the modular cocycle. This embeds the notion of unimodularity into a very general dynamical scheme of constructing and studying measures with a prescribed Radon–Nikodym cocycle.
A Functional CLT for Fields of Commuting Transformations Via Martingale Approximation
Аннотация
We consider a field f \( \circ {T}_1^{i_1}\circ \dots \circ {T}_d^{i_d} \) , where T1, . . . , Td are completely commuting transformations in the sense of Gordin. If one of these transformations is ergodic, we give sufficient conditions in the spirit of Hannan under which the partial sum process indexed by quadrants converges in distribution to a Brownian sheet. The proof combines a martingale approximation approach with a recent CLT for martingale random fields due to Volný. We apply our results to completely commuting endomorphisms of the m-torus. In that case, the conditions can be expressed in terms of the L2-modulus of continuity of f.
Kolmogorov Tests of Normality Based on Some Variants of Polya’s Characterization
Аннотация
Two variants of Kolmogorov-type U-empirical tests of normality are studied. They are based on variants of famous Polya’s characterization of the normal law. We calculate their local Bahadur efficiency against location, skew, and Lehmann alternatives and conclude that integral tests are usually more efficient.
On the Spectral Density of Stationary Processes and Random Fields
Аннотация
In this note, we show that a stationary sequence obtained by applying a fixed deterministic function to shifts of a stationary sequence (satisfying a mild regularity condition) has a spectral density. In the multiparametric setting, we obtain a similar result for a function of a shifted i.i.d. field.
Closability, Regularity, and Approximation by Graphs for Separable Bilinear Forms
Аннотация
We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact, separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e., energy forms on finite weighted graphs, in the sense of Mosco convergence, i.e., strong resolvent convergence.