arXiv subjects

A. A. Khartov

Publications and source records attributed to A. A. Khartov.

13 recordsLinked to original sources

arXiv2025

The class $\boldsymbol{Q}$ and mixture distributions with dominated continuous singular parts

We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A distribution function of the class $\boldsymbol{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy-type representation with a ``signed spectral measure''. This class is a wide natural extension of the fundamental class of infinitely divisible distribution functions and it is actively studied now. We are interested in conditions for a distribution function $F$ to belong to the class $\boldsymbol{Q}$ for the unexplored case, where $F$ may have a continuous singular part. We propose a criterion under the assumption that the continuous singular part of $F$ is dominated by the discrete part in a certain sense. The criterion generalizes the previous results by Alexeev and Khartov for discrete probability laws and the results by Berger and Kutlu for the mixtures of discrete and absolutely continuous laws. In addition, we describe the characteristic triplet of the corresponding Lévy-type representation, which may contain some continuous singular part. We also show that the assumption of the dominated continuous singular part cannot be simply omitted or even slightly extended (without some special assumptions). We apply the general criterion to some interesting particular examples. We also positively solve the decomposition problem stated by Lindner, Pan and Sato within the considered case.

math.PR↗

arXiv2024

On decomposition problem for distribution functions of class $\boldsymbol{Q}$

We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A distribution function of the class $\boldsymbol{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy--Khinchine type representation with a ``signed spectral measure''. The class $\boldsymbol{Q}$, being a natural extension of the class $\boldsymbol{I}$ of infinitely divisible distribution functions, is actively studied now and it finds various applications. In 2018, Lindner, Pan and Sato formulated the open question: is it true that if $F\in\boldsymbol{Q}$ and $F=F_1*F_2$ with some distribution functions $F_1$ and $F_2$, then $F_1\in\boldsymbol{Q}$ and $F_2\in\boldsymbol{Q}$? There are some positive results under special assumptions on the type of $F$. In this paper, we answer the question in a general setting without any additional assumptions. We also consider the same question but with the stronger assumption that $F\in\boldsymbol{I}$.

math.PR↗

arXiv2024

Some criteria of rational-infinite divisibility for probability laws

We study the class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. The class $\boldsymbol{Q}$ is a wide natural extension of the fundamental class of infinitely divisible distribution functions. We are interested in general conditions to belong to the class $\boldsymbol{Q}$ in terms of characteristic functions. We obtain criteria that seem to be convenient for the application for some cases, and we illustrate it by several examples in the paper.

math.PR↗

arXiv2023

Asymptotic analysis in multivariate worst case approximation with Gaussian kernels

We consider a problem of approximation of $d$-variate functions defined on $\mathbb{R}^d$ which belong to the Hilbert space with tensor product-type reproducing Gaussian kernel with constant shape parameter. Within worst case setting, we investigate the growth of the information complexity as $d\to\infty$. The asymptotics are obtained for the case of fixed error threshold and for the case when it goes to zero as $d\to\infty$.

math.PR↗

arXiv2023

A criterion and a Cramér-Wold device for quasi-infinite divisibility for discrete multivariate probability laws

Multivariate discrete probability laws are considered. We show that such laws are quasi-infinitely divisible if and only if their characteristic functions are separated from zero. We generalize the existing results for the univariate discrete laws and for the multivariate laws on $\mathbb{Z}^d$. The Cramér-Wold devices for infinite and quasi-infinite divisibility were proved.

math.PR↗

arXiv2022

On weak convergence of quasi-infinitely divisible laws

We study a new class of so-called quasi-infinitely divisible laws, which is a wide natural extension of the well known class of infinitely divisible laws through the Lévy--Khinchine type representations. We are interested in criteria of weak convergence within this class. Under rather natural assumptions, we state assertions, which connect a weak convergence of quasi-infinitely divisible distribution functions with one special type of convergence of their Lévy--Khinchine spectral functions. The latter convergence is not equivalent to the weak convergence. So we complement known results by Lindner, Pan, and Sato (2018) in this field.

math.PR↗

arXiv2021

A criterion of quasi-infinite divisibility for discrete laws

We consider arbitrary discrete probability laws on the real line. We obtain a criterion of their belonging to a new class of quasi-infinitely divisible laws, which is a wide natural extension of the class of well known infinitely divisible laws through the Lévy type representations.

math.PR↗

arXiv2021

Spectral representations of characteristic functions of discrete probability laws

We consider discrete probability laws on the real line, whose characteristic functions are separated from zero. In particular, this class includes arbitrary discrete infinitely divisible laws and lattice probability laws, whose characteristic functions have no zeroes on the real line. We show that characteristic functions of such laws admit spectral Lévy--Khinchine type representations. We also apply the representations of such laws to obtain limit and compactness theorems with convergence in variation to probability laws from this class. Thus we generalize the corresponding results from the recent papers by Lindner, Pan, Sato, and by Khartov.

math.PR↗

arXiv2021

Asymptotic analysis in multivariate average case approximation with Gaussian kernels

We consider tensor product random fields $Y_d$, $d\in\mathbb{N}$, whose covariance funtions are Gaussian kernels. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Namely, we find criteria of boundedness for $n^{Y_d}(\varepsilon)$ on $d$ and of tending $n^{Y_d}(\varepsilon)\to\infty$, $d\to\infty$, for any fixed $\varepsilon\in(0,1)$. In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics \begin{eqnarray*} \ln n^{Y_d}(\varepsilon)= a_d+q(\varepsilon)b_d+o(b_d),\quad d\to\infty, \end{eqnarray*} with any $\varepsilon\in(0,1)$. Here $q\colon (0,1)\to\mathbb{R}$ is a non-decreasing function, $(a_d)_{d\in\mathbb{N}}$ is a sequence and $(b_d)_{d\in\mathbb{N}}$ is a positive sequence such that $b_d\to\infty$, $d\to\infty$. We show that only special quantiles of self-decomposable distribution functions appear as functions $q$ in a given asymptotics. These general results apply to $n^{Y_d}(\varepsilon)$ under particular assumptions on the length scale parameters.

math.PR↗

arXiv2018

Approximation complexity of homogeneous sums of random processes

We study approximation properties of additive random fields $Y_d$, $d\in\mathbb{N}$, which are sums of zero-mean random processes with the same continuous covariance functions. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. The results are applied to sums of standard Wiener processes.

math.PR↗

arXiv2017

Asymptotic analysis of average case approximation complexity of additive random fields

We study approximation properties of sequences of centered additive random fields $Y_d$, $d\in\mathbb{N}$. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate $Y_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Under natural assumptions we obtain general results concerning asymptotics of $n^{Y_d}(\varepsilon)$. We apply our results to additive random fields with marginal random processes corresponding to the Korobov kernels.

math.PR↗

arXiv2015

Simplified criterion of quasi-polynomial tractability and its applications

We study approximation properties of sequences of centered random elements $X_d$, $d\in\mathbb{N}$, with values in separable Hilbert spaces. We focus on sequences of tensor product-type random elements, which have covariance operators of corresponding tensor product form. The average case approximation complexity $n^{X_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate $X_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. The growth of $n^{X_d}(\varepsilon)$ as a function of $\varepsilon^{-1}$ and $d$ determines whether a sequence of corresponding approximation problems for $X_d$, $d\in\mathbb{N}$, is tractable or not. Different types of tractability were studied in the paper by M. A. Lifshits, A. Papageorgiou and H. Woźniakowski (2012), where for each type the necessary and sufficient conditions were found in terms of the eigenvalues of the marginal covariance operators. We revise the criterion of quasi-polynomial tractability and provide its simplified version. We illustrate our result by applying to random elements corresponding to tensor products of squared exponential kernels. Also we extend recent result of G. Xu (2014) concerning weighted Korobov kernels.

math.PR↗

arXiv2014

Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements

We study approximation properties of sequences of centered random elements $X_d$, $d\in\mathbb N$, with values in separable Hilbert spaces. We focus on sequences of tensor product-type and, in particular, degree-type random elements, which have covariance operators of corresponding tensor form. The average case approximation complexity $n^{X_d}(\varepsilon)$ is defined as the minimal number of continuous linear functionals that is needed to approximate $X_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. In the paper we investigate $n^{X_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Namely, we find criteria of (un)boundedness for $n^{X_d}(\varepsilon)$ on $d$ and of tending $n^{X_d}(\varepsilon)\to\infty$, $d\to\infty$, for any fixed $\varepsilon\in(0,1)$. In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics \begin{eqnarray*} \ln n^{X_d}(\varepsilon)= a_d+q(\varepsilon)b_d+o(b_d),\quad d\to\infty, \end{eqnarray*} at continuity points of a non-decreasing function $q\colon (0,1)\to\mathbb R$. Here $(a_d)_{d\in\mathbb N}$ is a sequence and $(b_d)_{d\in\mathbb N}$ is a positive sequence such that $b_d\to\infty$, $d\to\infty$. Under rather weak assumptions, we show that for tensor product-type random elements only special quantiles of self-decomposable or, in particular, stable (for tensor degrees) probability distributions appear as functions $q$ in the asymptotics. We apply our results to the tensor products of the Euler integrated processes with a given variation of smoothness parameters and to the tensor degrees of random elements with regularly varying eigenvalues of covariance operator.

math.PR↗