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Weighted entropy: basic inequalities

Mark Kelbert Izabella Stuhl Yuri Suhov

https://doi.org/10.15559/17-MSTA85

Pub. online: 2 Oct 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 3 (2017), pp. 233–252

Abstract

This paper represents an extended version of an earlier note [10]. The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. We analyse analogs of the Fisher information inequality and entropy power inequality for the weighted entropy and discuss connections with weighted Lieb’s splitting inequality. The concepts of rates of the weighted entropy and information are also discussed.

Quantifying non-monotonicity of functions and the lack of positivity in signed measures

Youri Davydov Ričardas Zitikis

https://doi.org/10.15559/17-VMSTA84

Pub. online: 28 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 219–231

Abstract

In various research areas related to decision making, problems and their solutions frequently rely on certain functions being monotonic. In the case of non-monotonic functions, one would then wish to quantify their lack of monotonicity. In this paper we develop a method designed specifically for this task, including quantification of the lack of positivity, negativity, or sign-constancy in signed measures. We note relevant applications in Insurance, Finance, and Economics, and discuss some of them in detail.

Quantifying non-monotonicity of functions and the lack of positivity in signed measures

Youri Davydov Ričardas Zitikis

https://doi.org/10.15559/17-MSTA84

Pub. online: 28 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 3 (2017), pp. 219–231

Abstract

Cardiac intraventricular mass from lung adenocarcinoma

Daniel Grados-Saso Isaac Lacambra Ana Callejo Guillermo Pinillos Jara Gayán Francisco Cueva

https://doi.org/10.1515/semcard-2017-0002

Pub. online: 23 Sep 2017 Type: Case Report

Open Access

Journal: Seminars in Cardiovascular Medicine

Abstract

We present a sole intracardiac mass with no other cardiac involvement in a patient with metastatic lung cancer disease. This mass can be well characterized by advanced 3D echocardiography and echocardiographic contrast.

Asymptotic behavior of functionals of the solutions to inhomogeneous Itô stochastic differential equations with nonregular dependence on parameter

Grigorij Kulinich Svitlana Kushnirenko

https://doi.org/10.15559/17-VMSTA83

Pub. online: 22 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 199–217

Abstract

The asymptotic behavior, as $T\to \infty $, of some functionals of the form $I_{T}(t)=F_{T}(\xi _{T}(t))+{\int _{0}^{t}}g_{T}(\xi _{T}(s))\hspace{0.1667em}dW_{T}(s)$, $t\ge 0$ is studied. Here $\xi _{T}(t)$ is the solution to the time-inhomogeneous Itô stochastic differential equation

\[d\xi _{T}(t)=a_{T}\big(t,\xi _{T}(t)\big)\hspace{0.1667em}dt+dW_{T}(t),\hspace{1em}t\ge 0,\hspace{2.5pt}\xi _{T}(0)=x_{0},\]

$T>0$ is a parameter, $a_{T}(t,x),x\in \mathbb{R}$ are measurable functions, $|a_{T}(t,x)|\le C_{T}$ for all $x\in \mathbb{R}$ and $t\ge 0$, $W_{T}(t)$ are standard Wiener processes, $F_{T}(x),x\in \mathbb{R}$ are continuous functions, $g_{T}(x),x\in \mathbb{R}$ are measurable locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for $I_{T}(t)$ is established under nonregular dependence of $a_{T}(t,x)$ and $g_{T}(x)$ on the parameter T.

Asymptotic behavior of functionals of the solutions to inhomogeneous Itô stochastic differential equations with nonregular dependence on parameter

Grigorij Kulinich Svitlana Kushnirenko

https://doi.org/10.15559/17-MSTA83

Pub. online: 22 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 3 (2017), pp. 199–217

Abstract

\[d\xi _{T}(t)=a_{T}\big(t,\xi _{T}(t)\big)\hspace{0.1667em}dt+dW_{T}(t),\hspace{1em}t\ge 0,\hspace{2.5pt}\xi _{T}(0)=x_{0},\]

The self-normalized Donsker theorem revisited

Peter Parczewski

https://doi.org/10.15559/17-VMSTA82

Pub. online: 18 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 189–198

Abstract

We extend the Poincaré–Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $D([0,1])$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Csörgő et al. (2003). Some notes on spheres with respect to $\ell _{p}$-norms are given.

The self-normalized Donsker theorem revisited