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Modern Stochastics: Theory and Applications*

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Journals: Modern Stochastics: Theory and Applications*

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The laws of iterated and triple logarithms for extreme values of regenerative processes

Alexander Marynych Ivan Matsak

https://doi.org/10.15559/20-MSTA147

Pub. online: 17 Feb 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 7, Issue 1 (2020), pp. 61–78

Abstract

We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the lim sup and a law of the triple logarithm for the lim inf. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424–445]. We apply our results to several queuing systems and a birth and death process.

Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions

Yuriy Kozachenko Enzo Orsingher Lyudmyla Sakhno Olga Vasylyk

https://doi.org/10.15559/19-MSTA146

Pub. online: 17 Dec 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 7, Issue 1 (2020), pp. 79–96

Abstract

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable φ-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema for solutions. We present the examples of processes for which the assumptions of the general result are verified and bounds are written in the explicit form. The main result is also specified for the case of Gaussian initial condition.

Subject index

https://doi.org/10.15559/19-MSTA64SI

Pub. online: 12 Dec 2019 Type: Subject Index

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 4 (2019), pp. 523–527

2010 Mathematics Subject Classification index

https://doi.org/10.15559/19-MSTA64MI

Pub. online: 12 Dec 2019 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 4 (2019), pp. 519–522

Keywords index

https://doi.org/10.15559/19-MSTA64KI

Pub. online: 12 Dec 2019 Type: Keywords Index

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 4 (2019), pp. 517–518

Author index

https://doi.org/10.15559/19-MSTA64AI

Pub. online: 12 Dec 2019 Type: Author Index

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 4 (2019), p. 515

Jackknife covariance matrix estimation for observations from mixture

Rostyslav Maiboroda Olena Sugakova

https://doi.org/10.15559/19-MSTA145

Pub. online: 7 Nov 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 4 (2019), pp. 495–513

Abstract

A general jackknife estimator for the asymptotic covariance of moment estimators is considered in the case when the sample is taken from a mixture with varying concentrations of components. Consistency of the estimator is demonstrated. A fast algorithm for its calculation is described. The estimator is applied to construction of confidence sets for regression parameters in the linear regression with errors in variables. An application to sociological data analysis is considered.

BSDEs and log-utility maximization for Lévy processes

Paolo Di Tella Hans-Jürgen Engelbert

https://doi.org/10.15559/19-MSTA144

Pub. online: 28 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 4 (2019), pp. 479–494

Abstract

In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for Lévy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary Lévy process.

On a linear functional for infinitely divisible moving average random fields

Stefan Roth

https://doi.org/10.15559/19-MSTA143

Pub. online: 22 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 4 (2019), pp. 443–478

Abstract

Given a low-frequency sample of the infinitely divisible moving average random field $\{{\textstyle\int _{{\mathbb{R}^{d}}}}f(t-x)\Lambda (dx),\hspace{2.5pt}t\in {\mathbb{R}^{d}}\}$, in [13] we proposed an estimator $\widehat{u{v_{0}}}$ for the function $\mathbb{R}\ni x\mapsto u(x){v_{0}}(x)=(u{v_{0}})(x)$, with $u(x)=x$ and ${v_{0}}$ being the Lévy density of the integrator random measure Λ. In this paper, we study asymptotic properties of the linear functional ${L^{2}}(\mathbb{R})\ni v\mapsto {\left\langle v,\widehat{u{v_{0}}}\right\rangle _{{L^{2}}(\mathbb{R})}}$, if the (known) kernel function f has a compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.

On estimation of expectation of simultaneous renewal time of time-inhomogeneous Markov chains using dominating sequence

Vitaliy Golomoziy

https://doi.org/10.15559/19-MSTA138

Pub. online: 14 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 3 (2019), pp. 333–343

Abstract

The main subject of the study in this paper is the simultaneous renewal time for two time-inhomogeneous Markov chains which start with arbitrary initial distributions. By a simultaneous renewal we mean the first time of joint hitting the specific set C by both processes. Under the condition of existence a dominating sequence for both renewal sequences generated by the chains and non-lattice condition for renewal probabilities an upper bound for the expectation of the simultaneous renewal time is obtained.

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