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Randomly stopped sums with consistently varying distributions

Edita Kizinevič Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/16-VMSTA60

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 165–179

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution function of the random sum $S_{\eta }=\xi _{1}+\xi _{2}+\cdots +\xi _{\eta }$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

Randomly stopped sums with consistently varying distributions

Edita Kizinevič Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/16-MSTA60

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 3, Issue 2 (2016), pp. 165–179

Abstract

Large deviation principle for one-dimensional SDEs with discontinuous coefficients

Alexei Kulik Daryna Sobolieva

https://doi.org/10.15559/16-VMSTA57

Pub. online: 1 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 145–164

Abstract

We establish the large deviation principle for solutions of one-dimensional SDEs with discontinuous coefficients. The main statement is formulated in a form similar to the classical Wentzel–Freidlin theorem, but under the considerably weaker assumption that the coefficients have no discontinuities of the second kind.

Large deviation principle for one-dimensional SDEs with discontinuous coefficients

Alexei Kulik Daryna Sobolieva

https://doi.org/10.15559/16-MSTA57

Pub. online: 1 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 3, Issue 2 (2016), pp. 145–164

Abstract

Approximations for a solution to stochastic heat equation with stable noise

Larysa Pryhara Georgiy Shevchenko

https://doi.org/10.15559/16-VMSTA56

Pub. online: 30 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 133–144

Abstract

We consider a Cauchy problem for stochastic heat equation driven by a real harmonizable fractional stable process Z with Hurst parameter $H>1/2$ and stability index $\alpha >1$. It is shown that the approximations for its solution, which are defined by truncating the LePage series for Z, converge to the solution.

Approximations for a solution to stochastic heat equation with stable noise

Larysa Pryhara Georgiy Shevchenko

https://doi.org/10.15559/16-MSTA56

Pub. online: 30 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 3, Issue 2 (2016), pp. 133–144

Abstract

On fractal faithfulness and fine fractal properties of random variables with independent

Q^{*}

-digits

Muslem Ibragim Grygoriy Torbin

https://doi.org/10.15559/16-VMSTA55

Pub. online: 9 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 119–131

Abstract

We develop a new technique to prove the faithfulness of the Hausdorff–Besicovitch dimension calculation of the family $\varPhi ({Q}^{\ast })$ of cylinders generated by ${Q}^{\ast }$-expansion of real numbers. All known sufficient conditions for the family $\varPhi ({Q}^{\ast })$ to be faithful for the Hausdorff–Besicovitch dimension calculation use different restrictions on entries $q_{0k}$ and $q_{(s-1)k}$. We show that these restrictions are of purely technical nature and can be removed. Based on these new results, we study fine fractal properties of random variables with independent ${Q}^{\ast }$-digits.

On fractal faithfulness and fine fractal properties of random variables with independent

Q^{*}

-digits

Muslem Ibragim Grygoriy Torbin

https://doi.org/10.15559/16-MSTA55

Pub. online: 9 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 3, Issue 2 (2016), pp. 119–131

Abstract

Large deviations for drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process

Dmytro Marushkevych

https://doi.org/10.15559/16-VMSTA54

Pub. online: 7 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 107–117

Abstract

We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein–Uhlenbeck process driven by mixed fractional Brownian motion.

Large deviations for drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process

Dmytro Marushkevych

https://doi.org/10.15559/16-MSTA54

Pub. online: 7 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 3, Issue 2 (2016), pp. 107–117

Abstract

We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein–Uhlenbeck process driven by mixed fractional Brownian motion.

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