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On fractional regularity of distributions of functions in gaussian random variables
Fractional Calculus and Applied Analysis. 2019. Vol. 22. No. 5. P. 1249-1268.
We study fractional smoothness of measures on R^k, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption.
Kosov E., Математические заметки 2022 Т. 111 № 1 С. 67-79
In this paper, we study bounds for the total variation distance between distributions of second order polynomials in normal random variables provided that they essentially depend on at least three variables. Moreover, we partially extend some recent bounds for the Kolmogorov distance between the distributions of norms of Gaussian random vectors to the case of ...
Added: June 1, 2022
Bogachev V., Kolesnikov A., Теория вероятностей и ее применения 2005 № 50(1) С. 27-51
Показано, что для заданных равномерно выпуклой меры μ на R∞, эквивалентной своему сдвигу на вектор (1,0,0,...), и вероятностной меры ν, абсолютно непрерывной относительно μ, найдется борелевское отображение Т = пространства R∞, переводящее меру μ в v и имеющее вид Т(х) = х + F(x), где F принимает значения в l2. Более того, если мера μ ...
Added: March 23, 2011
Kosov E., Journal of Mathematical Analysis and Applications 2018 Vol. 462 No. 1 P. 390-406
We show that a measure on the real line, that is the image of a log-concave measure under a polynomial of degree d, possesses a density from the Nikolskii–Besov class of fractional order 1/d. This result is used to prove an estimate for the total variation distance between such measures in terms of the Fortet–Mourier ...
Added: July 20, 2018
Bogachev V., Kolesnikov A., Medvedev K. V., Математический сборник 2005 Т. 196 № 3 С. 3-30
Получено новое тождество для энтропии нелинейного образа меры на Rn, дающее известное неравенство Талаграна. Исследованы треугольные отображения в Rn и R∞, т.е. отображения T, у которых i-я координатная функция T
i зависит только от переменных x 1,…, xi. С помощью этих отображений дано положительное решение известной открытой проблемы о представимости всякой вероятностной меры ν, абсолютно непрерывной ...
Added: March 26, 2013
Bogachev V., Kosov E., Zelenov G., Transactions of the American Mathematical Society 2018 Vol. 370 No. 6 P. 4401-4432
We prove that the distribution density of any non-constant polynomial f(\xi _1,\xi _2,\ldots ) of degree d in independent standard Gaussian random variables \xi _i (possibly, in infinitely many variables) always belongs to the Nikolskii-Besov space B^{1/d}(R) of fractional order 1/d (and this order is best possible), and an analogous result holds for polynomial mappings ...
Added: July 20, 2018
Bogachev V., Malofeev I. I., Potential analysis 2016 Vol. 44 No. 4 P. 767-792
We propose a new construction of surface measures on infinite-dimensional spaces ...
Added: February 1, 2017
Bogachev V., Зеленов Г. И., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 461 № 1 С. 14-17
Получены новые оценки для расстояния по вариации для слабо сходящихся многомерных распределений ...
Added: November 15, 2017
Bogachev V., Kosov E., Popova S., Доклады Академии наук 2017 Т. 476 № 6 С. 609-613
Введены и исследованы классы Никольского-Бесова относительно гауссовских мер. ...
Added: November 1, 2017
Bogachev V., Providence : American Mathematical Society, 1998
This book presents a systematic exposition of the modern theory of Gaussian measures. The basic properties of finite and infinite dimensional Gaussian distributions, including their linear and nonlinear transformations, are discussed. The book is intended for graduate students and researchers in probability theory, mathematical statistics, functional analysis, and mathematical physics. It contains a lot of ...
Added: March 10, 2014
Ulyanov V. V., Теория вероятностей и ее применения 2015 Т. 60 № 2 С. 391-402
В работе рассматриваются различные свойства многочленов от случайных элементов: оценки характеристических функционалов многочленов, стохастическое обобщение теоремы Виноградова о среднем, характеризационная проблема, оценка вероятностей попадания в шары. При этом результаты охватывают случайные элементы со значениями как в конечномерных, так и в бесконечно-мерных гильбертовых пространствах. ...
Added: July 13, 2015
Kosov E., Doklady Mathematics 2019 Vol. 100 No. 2 P. 423-425
In the paper we discuss a new bound of the total variation distance in terms of L^2 distance for random variables that are polynominals in log-concave random vectors. ...
Added: November 16, 2019
Malinnikova E., Nikolay N. Osipov, Journal of Fourier Analysis and Applications 2019 Vol. 25 No. 3 P. 804-818
We discuss generalizations of Rubio de Francia’s inequality for Triebel–Lizorkin and Besov spaces, continuing the research from Osipov (Sb Math 205(7): 1004–1023, 2014) and answering Havin’s question to one of the authors. Two versions of Rubio de Francia’s operator are discussed: it is shown that exponential factors are needed for the boundedness of the operator ...
Added: October 30, 2018
Bogachev V., Providence : American Mathematical Society, 2010
The book gives a systematic account of the theory of differentiable measures and the Malliavin calculus. ...
Added: March 5, 2014
Bogachev V., Теория вероятностей и ее применения 2021 Т. 66 № 4 С. 693-717
A survey is given about Chebyshev-Hermit polynomials and distributions of polynomials in Gaussian random variables. ...
Added: October 29, 2021
Bogachev V., Pilipenko A. Y., Shaposhnikov A. V., Journal of Mathematical Analysis and Applications 2014 Vol. 419 No. 2 P. 1023-1044
We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space. ...
Added: January 3, 2015
Kolesnikov A., Kosov E., Theory of Stochastic Processes 2017 Vol. 22 No. 38 P. 47-61
Let γ be the standard Gaussian measure on Rn and let Pγ be the space of probability measures that are absolutely continuous with respect to γ. We study lower bounds for the functional Fγ(µ) = Ent(µ) − 1 2W2 2 (µ, ν), where µ ∈ Pγ, ν ∈ Pγ, Ent(µ) = R log µ γ ...
Added: August 21, 2018
Remizov I., Modeling and Analysis of Information Systems 2015 Vol. 22 No. 3 P. 337-355
A parabolic partial differential equation 𝑢′𝑡(𝑡, 𝑥) = 𝐿𝑢(𝑡, 𝑥) is considered, where 𝐿 is a linear second-order differential operator with time-independent coefficients, which may depend on 𝑥. We assume that the spatial coordinate 𝑥 belongs to a finite- or infinite-dimensional real separable Hilbert space 𝐻. Assuming the existence of a strongly continuous resolving semigroup ...
Added: October 30, 2018
Zelenov G., Theory of Stochastic Processes 2017 Vol. 22 No. 2 P. 79-85
We estimate total variation distances between distributions of polynomials viaL2-norms. ...
Added: November 30, 2019
Bogachev V., Kosov E., Popova S., Doklady Mathematics 2017 Vol. 96 No. 2 P. 498-502
A new characterization of Gaussian Besov-Nikolskii spaces is given. ...
Added: November 19, 2019
Bogachev V., Kosov E., Popova S., Izvestiya. Mathematics 2021 Vol. 85 No. 5 P. 852-882
New results on distributions of functionals on spaces with Gaussian measures are obtained. ...
Added: October 29, 2021
Bogachev V., Kosov E., Popova S., Moscow Mathematical Journal 2019 Vol. 19 No. 4 P. 619-654
A new approach to Nikolskii-Besov classes is presented. ...
Added: November 16, 2019
Bogachev V., Kolesnikov A., / Cornell University. Series math "arxiv.org". 2011. No. 1110.1822.
Given the standard Gaussian measure $\gamma$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g \cdot \gamma$ absolutely continuous with respect to $\gamma$, we consider the optimal transportation $T(x) = x + \nabla \varphi(x)$ of $g \cdot \gamma$ to $\gamma$. Assume that the function $|\nabla g|^2/g$ is $\gamma$-integrable. We prove that the ...
Added: March 28, 2013
Los A., Промышленные АСУ и контроллеры 2013 № 7 С. 31-36
В работе исследуется распределение суммы конечно-зависимых случайных величин, необходимость изучения которого возникает в ряде задач защиты информации. Одной из таких задач является исследование свойств выходной последовательности фильтрующего генератора, входящего в состав ряда отечественных изарубежных алгоритмов защиты информации. Полученные в статье оценки близости исследуемого распределения к распределению суммы независимых случайных величин позволяют сделать вывод о качестве ...
Added: March 11, 2015
Singapore : World Scientific, 2014
A survey of recent progress and open problems in the theory of Gaussian measures is given. ...
Added: January 3, 2015