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Convexity and robustness of the Rényi entropy

Filipp Buryak

Yuliya Mishura

https://doi.org/10.15559/21-VMSTA185

Pub. online: 26 Jul 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 387–412

Abstract

We study convexity properties of the Rényi entropy as function of $\alpha >0$ on finite alphabets. We also describe robustness of the Rényi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform but the number of events increases to ∞ and prove that the limit of Rényi entropy of the binomial distribution is equal to Rényi entropy of the Poisson distribution.

Estimation in a linear errors-in-variables model under a mixture of classical and Berkson errors

Mykyta Yakovliev

Alexander Kukush

https://doi.org/10.15559/21-VMSTA186

Pub. online: 26 Jul 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 373–386

Abstract

A linear structural regression model is studied, where the covariate is observed with a mixture of the classical and Berkson measurement errors. Both variances of the classical and Berkson errors are assumed known. Without normality assumptions, consistent estimators of model parameters are constructed and conditions for their asymptotic normality are given. The estimators are divided into two asymptotically independent groups.

Editorial

Mark Podolskij

https://doi.org/10.15559/21-VMSTA183

Pub. online: 23 Jun 2021 Type: Editorial

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 2 (2021), p. 139

Sharp asymptotics for q-norms of random vectors in high-dimensional

ℓ_{p}^{n}

-balls

Tom Kaufmann

https://doi.org/10.15559/21-VMSTA182

Pub. online: 22 Jun 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 2 (2021), pp. 239–274

Abstract

Sharp large deviation results of Bahadur–Ranga Rao type are provided for the q-norm of random vectors distributed on the ${\ell _{p}^{n}}$-ball ${\mathbb{B}_{p}^{n}}$ according to the cone probability measure or the uniform distribution for $1\le q<p<\infty $, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different ${\ell _{p}^{n}}$-balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an ${\ell _{p}^{n}}$-ball onto a line with uniform random direction. The sharp large deviation results are proven by providing convenient probabilistic representations of the q-norms, employing local limit theorems to approximate their densities, and then using geometric results for asymptotic expansions of Laplace integrals to integrate these densities and derive concrete probability estimates.

Second order elliptic partial differential equations driven by Lévy white noise

David Berger

Farid Mohamed

https://doi.org/10.15559/21-VMSTA181

Pub. online: 22 Jun 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 2 (2021), pp. 179–207

Abstract

This paper deals with linear stochastic partial differential equations with variable coefficients driven by Lévy white noise. First, an existence theorem for integral transforms of Lévy white noise is derived and the existence of generalized and mild solutions of second order elliptic partial differential equations is proved. Further, the generalized electric Schrödinger operator for different potential functions V is discussed.

Malliavin–Stein method: a survey of some recent developments

Ehsan Azmoodeh Giovanni Peccati Xiaochuan Yang

https://doi.org/10.15559/21-VMSTA184

Pub. online: 22 Jun 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 2 (2021), pp. 141–177

Abstract

Initiated around the year 2007, the Malliavin–Stein approach to probabilistic approximations combines Stein’s method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin–Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin–Stein method, with specific emphasis on extensions and generalizations in the framework of Markov semigroups and of random point measures.

Optimal transport between determinantal point processes and application to fast simulation

Laurent Decreusefond

Guillaume Moroz

https://doi.org/10.15559/21-VMSTA180

Pub. online: 2 Jun 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 2 (2021), pp. 209–237

Abstract

Two optimal transport problems between determinantal point processes (DPP for short) are investigated. It is shown how to estimate the Kantorovitch–Rubinstein and Wasserstein-2 distances between distributions of DPP. These results are applied to evaluate the accuracy of a fast but approximate simulation algorithm of the Ginibre point process restricted to a circle. One can now simulate in a reasonable amount of time more than ten thousands points.

Metatimes, random measures and cylindrical random variables

Fred Espen Benth

Iben Cathrine Simonsen

https://doi.org/10.15559/21-VMSTA178

Pub. online: 20 May 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 349–371

Abstract

Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.

Convergence rate of CLT for the drift estimation of sub-fractional Ornstein–Uhlenbeck process of second kind

Maoudo Faramba Baldé Khalifa Es-Sebaiy

https://doi.org/10.15559/21-VMSTA179

Pub. online: 20 May 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 329–347

Abstract

In this paper, we deal with an Ornstein–Uhlenbeck process driven by sub-fractional Brownian motion of the second kind with Hurst index $H\in (\frac{1}{2},1)$. We provide a least squares estimator (LSE) of the drift parameter based on continuous-time observations. The strong consistency and the upper bound $O(1/\sqrt{n})$ in Kolmogorov distance for central limit theorem of the LSE are obtained. We use a Malliavin–Stein approach for normal approximations.

On nonnegative solutions of SDDEs with an application to CARMA processes

Mikkel Slot Nielsen

Victor Rohde

https://doi.org/10.15559/21-VMSTA177

Pub. online: 31 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 309–328

Abstract

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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