Latest articles of Modern Stochastics: Theory and Applications
http://vmsta.org/journal/VMSTA/feeds/latest
https://vmsta.org/https://vmsta.org/Latest articles of Modern Stochastics: Theory and Applications
http://vmsta.org/journal/VMSTA/feeds/latest
enSun, 04 Jun 2023 08:10:23 +0300<![CDATA[On geometric recurrence for time-inhomogeneous autoregression]]>
https://vmsta.org/journal/VMSTA/article/271
https://vmsta.org/journal/VMSTA/article/271The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form ${\mathit{X}}_{\mathit{n}+1}={\mathit{\alpha}}_{\mathit{n}}{\mathit{X}}_{\mathit{n}}+{\mathit{\epsilon}}_{\mathit{n}}$, where ${\mathit{\alpha}}_{\mathit{n}}$ are constants, and ${\mathit{\epsilon}}_{\mathit{n}}$ are independent random variables. Conditions on ${\mathit{\alpha}}_{\mathit{n}}$ and distributions of ${\mathit{\epsilon}}_{\mathit{n}}$ are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes ${\mathit{X}}^{\mathrm{(}1\mathrm{)}}$ and ${\mathit{X}}^{\mathrm{(}2\mathrm{)}}$ assuming that both ${\mathit{\alpha}}_{\mathit{n}}^{\mathrm{(}\mathit{i}\mathrm{)}}$, $\mathit{i}\in \{1\mathrm{,}2\}$, and distributions of ${\mathit{\epsilon}}_{\mathit{n}}^{\mathrm{(}\mathit{i}\mathrm{)}}$, $\mathit{i}\in \{1\mathrm{,}2\}$, are close enough.
PDFXML]]>The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form ${\mathit{X}}_{\mathit{n}+1}={\mathit{\alpha}}_{\mathit{n}}{\mathit{X}}_{\mathit{n}}+{\mathit{\epsilon}}_{\mathit{n}}$, where ${\mathit{\alpha}}_{\mathit{n}}$ are constants, and ${\mathit{\epsilon}}_{\mathit{n}}$ are independent random variables. Conditions on ${\mathit{\alpha}}_{\mathit{n}}$ and distributions of ${\mathit{\epsilon}}_{\mathit{n}}$ are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes ${\mathit{X}}^{\mathrm{(}1\mathrm{)}}$ and ${\mathit{X}}^{\mathrm{(}2\mathrm{)}}$ assuming that both ${\mathit{\alpha}}_{\mathit{n}}^{\mathrm{(}\mathit{i}\mathrm{)}}$, $\mathit{i}\in \{1\mathrm{,}2\}$, and distributions of ${\mathit{\epsilon}}_{\mathit{n}}^{\mathrm{(}\mathit{i}\mathrm{)}}$, $\mathit{i}\in \{1\mathrm{,}2\}$, are close enough.
PDFXML]]>Vitaliy GolomoziyWed, 03 May 2023 00:00:00 +0300<![CDATA[On some composite Kies families: distributional properties and saturation in Hausdorff sense]]>
https://vmsta.org/journal/VMSTA/article/269
https://vmsta.org/journal/VMSTA/article/269The stochastic literature contains several extensions of the exponential distribution which increase its applicability and flexibility. In the present article, some properties of a new power modified exponential family with an original Kies correction are discussed. This family is defined as a Kies distribution which domain is transformed by another Kies distribution. Its probabilistic properties are investigated and some limitations for the saturation in the Hausdorff sense are derived. Moreover, a formula of a semiclosed form is obtained for this saturation. Also the tail behavior of these distributions is examined considering three different criteria inspired by the financial markets, namely, the VaR, AVaR, and expectile based VaR. Some numerical experiments are provided, too.
PDFXML]]>The stochastic literature contains several extensions of the exponential distribution which increase its applicability and flexibility. In the present article, some properties of a new power modified exponential family with an original Kies correction are discussed. This family is defined as a Kies distribution which domain is transformed by another Kies distribution. Its probabilistic properties are investigated and some limitations for the saturation in the Hausdorff sense are derived. Moreover, a formula of a semiclosed form is obtained for this saturation. Also the tail behavior of these distributions is examined considering three different criteria inspired by the financial markets, namely, the VaR, AVaR, and expectile based VaR. Some numerical experiments are provided, too.
PDFXML]]>Tsvetelin Zaevski,Nikolay KyurkchievTue, 21 Mar 2023 00:00:00 +0200<![CDATA[Consistency of LSE for the many-dimensional symmetric textured surface parameters]]>
https://vmsta.org/journal/VMSTA/article/268
https://vmsta.org/journal/VMSTA/article/268A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${\mathbb{R}^{M}},M\ge 3$.
PDFXML]]>A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${\mathbb{R}^{M}},M\ge 3$.
PDFXML]]>Oleksandr Dykyi,Alexander IvanovThu, 16 Mar 2023 00:00:00 +0200<![CDATA[Ruin probabilities as functions of the roots of a polynomial]]>
https://vmsta.org/journal/VMSTA/article/267
https://vmsta.org/journal/VMSTA/article/267A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.
PDFXML]]>A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.
PDFXML]]>David J. Santana,Luis RincónWed, 15 Mar 2023 00:00:00 +0200<![CDATA[The Burgers equation driven by a stochastic measure]]>
https://vmsta.org/journal/VMSTA/article/266
https://vmsta.org/journal/VMSTA/article/266The class of one-dimensional equations driven by a stochastic measure μ is studied. For μ only σ-additivity in probability is assumed. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.
PDFXML]]>The class of one-dimensional equations driven by a stochastic measure μ is studied. For μ only σ-additivity in probability is assumed. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.
PDFXML]]>Vadym RadchenkoTue, 21 Feb 2023 00:00:00 +0200<![CDATA[Bernstein-type bounds for beta distribution]]>
https://vmsta.org/journal/VMSTA/article/265
https://vmsta.org/journal/VMSTA/article/265This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-Gaussian and sub-gamma bounds previously studied in this context.

The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.

PDFXML]]>This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-Gaussian and sub-gamma bounds previously studied in this context.

The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.

PDFXML]]>Maciej SkorskiMon, 13 Feb 2023 00:00:00 +0200<![CDATA[Transport equation driven by a stochastic measure]]>
https://vmsta.org/journal/VMSTA/article/264
https://vmsta.org/journal/VMSTA/article/264The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.
PDFXML]]>The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.
PDFXML]]>Vadym RadchenkoMon, 06 Feb 2023 00:00:00 +0200<![CDATA[Parameter estimation in mixed fractional stochastic heat equation]]>
https://vmsta.org/journal/VMSTA/article/263
https://vmsta.org/journal/VMSTA/article/263The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3>. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.
PDFXML]]>The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3>. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.
PDFXML]]>Diana Avetisian,Kostiantyn RalchenkoTue, 24 Jan 2023 00:00:00 +0200<![CDATA[A modified Φ-Sobolev inequality for canonical Lévy processes and its applications]]>
https://vmsta.org/journal/VMSTA/article/262
https://vmsta.org/journal/VMSTA/article/262A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.
PDFXML]]>A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.
PDFXML]]>Noriyoshi Sakuma,Ryoichi SuzukiMon, 23 Jan 2023 00:00:00 +0200<![CDATA[Some examples of noncentral moderate deviations for sequences of real random variables]]>
https://vmsta.org/journal/VMSTA/article/261
https://vmsta.org/journal/VMSTA/article/261The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.
PDFXML]]>The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.
PDFXML]]>Rita Giuliano,Claudio MacciThu, 19 Jan 2023 00:00:00 +0200