arXiv subjects

Viacheslav V. Saenko

Publications and source records attributed to Viacheslav V. Saenko.

8 recordsLinked to original sources

arXiv2023

The calculation of the distribution function of a strictly stable law at large X

The paper considers the problem of calculating the distribution function of a strictly stable law at $x\to\infty$. To solve this problem, an expansion of the distribution function in a power series was obtained, and an estimate of the remainder term was also obtained. It was shown that in the case $α<1$ this series was convergent for any $x$, in the case $α=1$ the series was convergent at $N\to\infty$ in the domain $|x|>1$, and in the case $α>1$ the series was asymptotic at $x\to\infty$. The case $α=1$ was considered separately and it was demonstrated that in that case the series converges to the generalized Cauchy distribution. An estimate for the threshold coordinate $x_\varepsilon^N$ was obtained which determined the area of applicability of the obtained expansion. It was shown that in the domain $|x|\geqslant x_\varepsilon^N$ this power series could be used to calculate the distribution function, which completely solved the problem of calculating the distribution function at large $x$.

math.ST↗

arXiv2023

The calculation of the probability density of a strictly stable law at large $X$

The article is devoted to the problem of calculating the probability density of a strictly stable law at $x\to\infty$. To solve this problem, it was proposed to use the expansion of the probability density in a power series. A representation of the probability density in the form of a power series and an estimate for the remainder term was obtained. This power series is convergent in the case $0<α<1$ and asymptotic at $x\to\infty$ in the case $1<α<2$. The case $α=1$ was considered separately. It was shown that in the case $α=1$ the obtained power series was convergent for any $|x|>1$ at $N\to\infty$. It was also shown that in this case it was convergent to the density of $g(x,1,θ)$. An estimate of the threshold coordinate $x_\varepsilon^N$, was obtained which determines the range of applicability of the resulting expansion of the probability density in a power series. It was shown that in the domain $|x|\geqslant x_\varepsilon^N$ this power series could be used to calculate the probability density.

math.PR↗

arXiv2022

The calculation of the probability density and distribution function of a strictly stable law in the vicinity of zero

The problem of calculating the probability density and distribution function of a strictly stable law is considered at $x\to0$. The expansions of these values into power series were obtained to solve this problem. It was shown that in the case $α<1$ the obtained series were asymptotic at $x\to0$, in the case $α>1$ they were convergent and in the case $α=1$ in the domain $|x|<1$ these series converged to an asymmetric Cauchy distribution. It has been shown that at $x\to0$ the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly stable laws.

math.ST↗

arXiv2021

Integral representation of the Mittag-Leffler function

Generalization of the integral representation of the gamma function has been obtained, which shows that the Hankel contour assumes rotation in the complex plane. The range of admissible values for the contour rotation angle is set. Using this integral representation, generalization of the integral representation of the Mittag-Leffler function has been obtained that expresses the value of this function in terms of the contour integral.

math.CA↗

arXiv2021

The calculation of the Mittag-Leffler function

The problem of calculating the Mittag-Leffler function $E_{ρ,μ} (z)$ is considered in the paper. To solve this problem integral representations for the function $E_{ρ,μ}(z)$ are transformed in such a way that they could not contain complex variables and parameters. Integral representations written in this form allow one to use standard methods of numerical integration to calculate integrals contained in them. To verify the correctness of the integral representations obtained the function $E_{ρ,μ}(z)$ was calculated both with the use of obtained formulas and with the use of known representations of the Mittag-Leffler function. The calculation results demonstrate their exact matching. This fact is indicative of the correctness of new integral representations of the function $E_{ρ,μ}(z)$ that were obtained.

math.CA↗

arXiv2020

Two forms of the integral representations of the Mittag-Leffler function

The integral representation of the two-parameter Mittag-Leffler function $E_{ρ,μ}(z)$ is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function $E_{ρ,μ}(z)$ has two forms: the representation ``A'' and ``B''. Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and the advantages and disadvantages of each of the obtained representations are discussed.

math.CA↗

arXiv2020

Singular points of the integral representation of the Mittag-Leffler function

The paper presents an integral representation of the two-parameter Mittag-Leffler function $E_{ρ,μ}(z)$ and singular points of this representation have been studied. It has been found that there are two singular points for this integral representation: $ζ=1$ and $ζ=0$. The point $ζ=1$ is a pole of the first order and the point $ζ=0$, depending on the values of parameters $ρ,μ$ is either a pole or a branch point, or a regular point. The subsequent study showed that at some values of parameters $ρ,μ$ with the help of the residue theory one can calculate the integral included in the studied integral representation and express the function $E_{ρ,μ}(z)$ through elementary functions.

math.CA↗

arXiv2015

The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model

Levy walk at the finite velocity is considered. To analyze the spatial and temporal characteristics of this process, the method of moments has been used. The asymptotic distributions of the moments (at $t\to\infty$) have been obtained for $N$ dimensional case where the free path of particles demonstrates the power-law distribution $p_ξ(x)=αx_0^αx^{-α-1}$, $x\to\infty$, $0<α<2$. The three regimes of distribution have been distinguished: ballistic, diffusion and asymptotic. Introduction of the finite velocity requires considering of two problems: propagation with distribution at the finite mathematical expectation of the free path ($1<α<2$) and propagation with distribution at the infinite mathematical expectation of the free path of the particle ($0<α<1$). In the case $1<α<2$, the asymptotic distribution is described by the Levy stable law and the effect of the finite velocity is reduced to a decrease of diffusivity. At $0<α<1$, the situation is quite different. Here, the asymptotic distribution exhibits a $U$- or $W$-shape and is described as the ballistic regime of distribution. The obtained moments allow to reconstruct the distribution densities of particles in one-dimensional and three-dimensional cases.

astro-ph.GA↗