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On generalized stochastic fractional integrals and related inequalities

Hüseyin Budak Mehmet Zeki Sarikaya

https://doi.org/10.15559/18-MSTA117

Pub. online: 24 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 471–481

Abstract

The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals.

Stochastic models associated to a Nonlocal Porous Medium Equation

Alessandro De Gregorio

https://doi.org/10.15559/18-VMSTA112

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 457–470

Abstract

The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights.

Stochastic models associated to a Nonlocal Porous Medium Equation

Alessandro De Gregorio

https://doi.org/10.15559/18-MSTA112

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 457–470

Abstract

Drifted Brownian motions governed by fractional tempered derivatives

Mirko D’Ovidio Francesco Iafrate Enzo Orsingher

https://doi.org/10.15559/18-VMSTA114

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 445–456

Abstract

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann–Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.

Drifted Brownian motions governed by fractional tempered derivatives

Mirko D’Ovidio Francesco Iafrate Enzo Orsingher

https://doi.org/10.15559/18-MSTA114

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 445–456

Abstract

Approximation of solutions of the stochastic wave equation by using the Fourier series

Vadym Radchenko

Nelia Stefans’ka

https://doi.org/10.15559/18-VMSTA115

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 429–444

Abstract

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fejèr sums we obtain the approximations of mild solution of the equation.

Approximation of solutions of the stochastic wave equation by using the Fourier series

Vadym Radchenko

Nelia Stefans’ka

https://doi.org/10.15559/18-MSTA115

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 429–444

Abstract

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

Björn Böttcher Martin Keller-Ressel René L. Schilling

https://doi.org/10.15559/18-VMSTA116

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 353–383

Abstract

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework.

From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the present article.

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

Björn Böttcher Martin Keller-Ressel René L. Schilling

https://doi.org/10.15559/18-MSTA116

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 3 (2018), pp. 353–383

Abstract

On a bound of the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables

Anatolii Zolotukhin Sergei Nagaev Vladimir Chebotarev

https://doi.org/10.15559/18-VMSTA113

Pub. online: 14 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 385–410

Abstract

It is shown that the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\le n\le 500000$, where n is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009–2011, an upper bound is obtained for the absolute constant in the Berry–Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.06%. As an auxiliary result, we prove a bound in the local Moivre–Laplace theorem which has a simple and explicit form.

Despite the best possible result, obtained by J. Schulz in 2016, we propose our approach to the problem of finding the absolute constant in the Berry–Esseen inequality for two-point distributions since this approach, combining analytical methods and the use of computers, could be useful in solving other mathematical problems.

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