Recent zbMATH articles in MSC 60Ehttps://zbmath.org/atom/cc/60E2024-03-13T18:33:02.981707ZWerkzeugSharp growth of the Ornstein-Uhlenbeck operator on Gaussian tail spaceshttps://zbmath.org/1528.410252024-03-13T18:33:02.981707Z"Eskenazis, Alexandros"https://zbmath.org/authors/?q=ai:eskenazis.alexandros"Ivanisvili, Paata"https://zbmath.org/authors/?q=ai:ivanisvili.paataSummary: Let \(X\) be a standard Gaussian random variable. For any \(p \in (1, \infty)\), we prove the existence of a universal constant \(C_p> 0\) such that the inequality
\[
(\mathbb{E} | h^\prime (X) |^p)^{1/p} \geq C_p \sqrt{d} (\mathbb{E} | h(X)|^p)^{1/p}
\]
holds for all \(d \geq 1\) and all polynomials \(h : \mathbb{R} \to \mathbb{C}\) whose spectrum is supported on frequencies at least \(d\), that is, \(\mathbb{E}h (X) X^k = 0\) for all \(k = 0, 1, \dots, d-1\). As an application of this optimal estimate, we obtain an affirmative answer to the Gaussian analogue of a question of \textit{M. Mendel} and \textit{A. Naor} [Publ. Math., Inst. Hautes Étud. Sci. 119, 1--95 (2014; Zbl 1306.46021)] concerning the growth of the Ornstein-Uhlenbeck operator on tail spaces of the real line. We also show the same bound for the gradient of analytic polynomials in an arbitrary dimension.Log concavity preservation by beta operator based on probability toolshttps://zbmath.org/1528.410422024-03-13T18:33:02.981707Z"Badía, F. G."https://zbmath.org/authors/?q=ai:badia.francisco-german"Lee, H."https://zbmath.org/authors/?q=ai:lee.hyunju"Sangüesa, C."https://zbmath.org/authors/?q=ai:sanguesa.carmenSummary: A number of papers have dealt with the preservation of log convexity and log concavity based on various operators. For instance, in [\textit{F. G. Badía} and \textit{C. Sangüesa}, J. Math. Anal. Appl. 413, No. 2, 953--962 (2014; Zbl 1310.41012)], the preservation of log convexity and log concavity under Bernstein operators was discussed based on some characteristics of a stochastic process. However, regarding beta-type operators, the preservation of log concavity does not hold based on such a probabilistic method used in [loc. cit.]. In this study, we explore the preservation of log concavity for the beta operator using alternative probabilistic tools. Notably, we show results on the preservation of log concavity for monotone log concave functions. Further, some results of application to some specific functions, ageing classes of the deterioration Dirichlet process, related operators and order statistics are provided.Polynomial approximations in a generalized Nyman-Beurling criterionhttps://zbmath.org/1528.410472024-03-13T18:33:02.981707Z"Alouges, François"https://zbmath.org/authors/?q=ai:alouges.francois"Darses, Sébastien"https://zbmath.org/authors/?q=ai:darses.sebastien"Hillion, Erwan"https://zbmath.org/authors/?q=ai:hillion.erwanSummary: The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on \((0, \infty)\), involving dilations of the fractional part function by factors \(\theta_k\in(0, 1)\), \(k \geq 1\). Randomizing the \(\theta_k\) gener
By the way, I had editing rights over that page, maybe they should be
transferred to you. The person I was in touch with was '''Robert
Gieseke''' <>ates new structures and criteria. One of them is a sufficient condition for RH that splits into
\begin{itemize}
\item[(i)] showing that the indicator function can be approximated by convolution with the fractional part,
\item[(ii)] a control on the coefficients of the approximation.
\end{itemize}
This self-contained paper generalizes conditions (i) and (ii) that involve a \(\sigma_0\in(1/2, 1)\), and imply \(\zeta (\sigma +it) \neq 0\) in the strip \(1/2 < \sigma \leq \sigma_0 < 1\). We then identify functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In this context, the difficulty for proving RH is then reallocated in (ii), which heavily relies on the corresponding Gram matrices, for which two remarkable structures are obtained. We show that a particular tuning of the approximating sequence leads to a striking simplification of the second Gram matrix, then reading as a block Hankel form.The basic distributional theory for the product of zero mean correlated normal random variableshttps://zbmath.org/1528.600172024-03-13T18:33:02.981707Z"Gaunt, Robert E."https://zbmath.org/authors/?q=ai:gaunt.robert-edwardSummary: The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener-Itô integrals.
{{\copyright} 2022 The Author. \textit{Statistica Neerlandica} published by John Wiley \& Sons Ltd on behalf of Netherlands Society for Statistics and Operations Research.}On quantile-based asymmetric family of distributions: properties and inferencehttps://zbmath.org/1528.600182024-03-13T18:33:02.981707Z"Gijbels, Irène"https://zbmath.org/authors/?q=ai:gijbels.irene"Karim, Rezaul"https://zbmath.org/authors/?q=ai:karim.rezaul"Verhasselt, Anneleen"https://zbmath.org/authors/?q=ai:verhasselt.anneleenSummary: In this paper, we provide a detailed study of a general family of asymmetric densities. In the general framework, we establish expressions for important characteristics of the distributions and discuss estimation of the parameters via method-of-moments as well as maximum likelihood estimation. Asymptotic normality results for the estimators are provided. The results under the general framework are then applied to some specific examples of asymmetric densities. The use of the asymmetric densities is illustrated in a real-data analysis.
{{\copyright} 2019 The Authors. International Statistical Review {\copyright} 2019 International Statistical Institute}Response to the letter to the editor on ``On quantile-based asymmetric family of distributions: properties and inference''https://zbmath.org/1528.600192024-03-13T18:33:02.981707Z"Gijbels, Irène"https://zbmath.org/authors/?q=ai:gijbels.irene"Karim, Rezaul"https://zbmath.org/authors/?q=ai:karim.rezaul"Verhasselt, Anneleen"https://zbmath.org/authors/?q=ai:verhasselt.anneleenSummary: \textit{F. J. R. Alvarez} [Int. Stat. Rev. 88, No. 3, 793--796 (2020; Zbl 1528.60021)] points out an identification problem for the four-parameter family of two-piece asymmetric densities introduced by \textit{V. Nassiri} and \textit{I. Loris} [J. Appl. Stat. 40, No. 5, 1090--1105 (2013; Zbl 1514.62770)]. This implies that statistical inference for that family is problematic. Establishing probabilistic properties for this four-parameter family however still makes sense. For the three-parameter family, there is no identification problem. The main contribution in [the authors, ibid. 87, No. 3, 471--504 (2019; Zbl 1528.60018)] is to provide asymptotic results for maximum likelihood and method-of-moments estimators for members of the three-parameter quantile-based asymmetric family of distributions.
{{\copyright} 2020 International Statistical Institute}On recursions for moments of a compound random variable: an approach using an auxiliary counting random variablehttps://zbmath.org/1528.600202024-03-13T18:33:02.981707Z"Kim, Yoora"https://zbmath.org/authors/?q=ai:kim.yooraSummary: We present an identity on moments of a compound random variable by using an auxiliary counting random variable. Based on this identity, we develop a new recurrence formula for obtaining the raw and central moments of any order for a given compound random variable.Letter to the editor: ``On quantile-based asymmetric family of distributions: properties and inference''https://zbmath.org/1528.600212024-03-13T18:33:02.981707Z"Rubio Alvarez, Francisco J."https://zbmath.org/authors/?q=ai:alvarez.francisco-j-rubioSummary: We show that the family of asymmetric distributions studied in a recent publication in the International Statistical Review is equivalent to the family of two-piece distributions. Moreover, we show that the location-scale asymmetric family proposed in that publication is non-identifiable (overparameterised), and it coincides with the family of two-piece distributions after removing the redundant parameters.
{{\copyright} 2020 International Statistical Institute}
Comment to the paper [\textit{I. Gijbels} et al., Int. Stat. Rev. 87, No. 3, 471--504 (2019; Zbl 1528.60018)].Joint occupation times in an infinite interval for spectrally negative Lévy processes on the last exit timehttps://zbmath.org/1528.600222024-03-13T18:33:02.981707Z"Li, Yingqiu"https://zbmath.org/authors/?q=ai:li.yingqiu"Wei, Yushao"https://zbmath.org/authors/?q=ai:wei.yushao"Hu, Yangli"https://zbmath.org/authors/?q=ai:hu.yangliLévy processes are a kind of processes with stationary independent increments and play a very important role in many fields. A spectrally negative Lévy process (SNLP) is a Lévy process without positive jumps. With the Markov property and infinite divisibility, it plays an important role in the study of financial risks. The occupation time of an SNLP plays an important role in the study of bankruptcy problems. The authors of this article use the Poisson approach and perturbation approach to find some joint Laplace transforms of the last exit time by their joint occupation times over semiinfinite intervals \((-\infty,0)\) and \((0,\infty)\) by virtue of the associated scale functions.
Reviewer: Ze-Chun Hu (Chengdu)Azuma-Hoeffding bounds for a class of urn modelshttps://zbmath.org/1528.600262024-03-13T18:33:02.981707Z"Dasgupta, Amites"https://zbmath.org/authors/?q=ai:dasgupta.amitesSummary: We obtain Azuma-Hoeffding bounds, which are exponentially decreasing, for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using cyclic vectors. Moreover, strong convergence of proportions is proved as an application of these bounds.One dimensional martingale rearrangement couplingshttps://zbmath.org/1528.600342024-03-13T18:33:02.981707Z"Jourdain, B."https://zbmath.org/authors/?q=ai:jourdain.benjamin"Margheriti, W."https://zbmath.org/authors/?q=ai:margheriti.williamLet \(\mu\) and \(\nu\) be probability measures on \(\mathbb{R}\) with finite moment of order \(\rho\) for some \(\rho\geq 1\). Denote by \(\Pi(\mu,\nu)\) the set of couplings of \(\mu\) and \(\nu\), that is, the set of measures on \(\mathbb{R}^2\) with marginals \(\mu\) and \(\nu\). Also, let \(\Pi^{\mathrm{M}}(\mu,\nu)\) denote the set of martingale couplings of \(\mu\) and \(\nu\), that is, the set of \(M\in \Pi(\mu,\nu)\) with \(\int_{\mathbb{R}}y\frac{M(\mathrm{d}x, \mathrm{d}y)}{\mu(\mathrm{d}x)}=x\) \(\mu\) a.e.
Fix a coupling \(\pi\in \Pi(\mu,\nu)\) satisfying \(\int_{\mathbb{R}}f(x)\mu(\mathrm{d}x)\leq \int_{\mathbb{R}}f(y)\nu(\mathrm{d}y)\) for any convex function \(f:\mathbb{R}\to\mathbb{R}\) and \(\int_{[a,\infty)}(x-\int_{\mathbb{R}}y\frac{\pi(\mathrm{d}x, \mathrm{d}y)}{\mu(\mathrm{d}x)})\mu(\mathrm{d}x)\leq 0\) for any \(a\in\mathbb{R}\). The latter condition is called the barycentre dispersion assumption. The authors provide a martingale coupling \(M\in \Pi^{\mathrm{M}}(\mu, \nu)\) such that \(\mathcal{AW}_\rho(\pi, M)=\inf_{M^\prime\in \Pi^{\mathrm{M}}(\mu, \nu)}\mathcal{AW}_\rho(\pi, M^\prime)\), where \(\mathcal{AW}_\rho\) is the so called adapted Wasserstein distance.
Finally, the authors discuss stability in the adapted Wassertein distance of the inverse transform martingale coupling with respect to the marginals.
Reviewer: Alexander Iksanov (Kyïv)An optimal transport-based characterization of convex orderhttps://zbmath.org/1528.600362024-03-13T18:33:02.981707Z"Wiesel, Johannes"https://zbmath.org/authors/?q=ai:wiesel.johannes-c-w"Zhang, Erica"https://zbmath.org/authors/?q=ai:zhang.ericaSummary: For probability measures \(\mu\), \(\nu\), and \(\rho\), define the cost functionals
\[
C(\mu, \rho) := \sup_{\pi\in\Pi(\mu, \rho)}\int\langle x, y\rangle \pi(\mathrm{d}x, \mathrm{d}y) \text{ and } C(\nu, \rho) := \sup_{\pi\in\Pi(\nu, \rho)}\int\langle x, y\rangle\pi(\mathrm{d}x, \mathrm{d}y),
\]
where \(\langle\cdot, \cdot \rangle\) denotes the scalar product and \(\Pi(\cdot, \cdot)\) is the set of couplings. We show that two probability measures \(\mu\) and \(\nu\) on \(\mathbb{R}^d\) with finite first moments are in convex order (i.e., \(\mu\preccurlyeq_c\nu)\) iff \(C(\mu, \rho) \leq C(\nu, \rho)\) holds for all probability measures \(\rho\) on \(\mathbb{R}^d\) with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of \(\int f\mathrm{d}\nu - \int f\mathrm{d}\mu\) over all 1-Lipschitz functions \(f\), which is obtained through optimal transport (OT) duality and the characterization result of OT (couplings) by Rüschendorf, by Rachev, and by Brenier. Building on this result, we derive new proofs of well known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.Rudin extension theorems on product spaces, turning bands, and random fields on balls cross timehttps://zbmath.org/1528.600522024-03-13T18:33:02.981707Z"Porcu, Emilio"https://zbmath.org/authors/?q=ai:porcu.emilio"Feng, Samuel F."https://zbmath.org/authors/?q=ai:feng.samuel-f"Emery, Xavier"https://zbmath.org/authors/?q=ai:emery.xavier"Peron, Ana P."https://zbmath.org/authors/?q=ai:peron.ana-paulaSummary: Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into \(d\)-dimensional Euclidean spaces, \(\mathbb{R}^d\), have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggests that the problem is challenging. This paper provides extension theorems for multiradial characteristic functions that are defined in balls embedded in \(\mathbb{R}^d\) cross, either \(\mathbb{R}^{d'}\) or the unit sphere \(\mathbb{S}^{d'}\) embedded in \(\mathbb{R}^{d'+1}\), for any two positive integers \(d\) and \(d'\). We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.Stochastic comparisons of relevation allocation policies in coherent systemshttps://zbmath.org/1528.900982024-03-13T18:33:02.981707Z"Zhang, Jiandong"https://zbmath.org/authors/?q=ai:zhang.jiandong"Zhang, Yiying"https://zbmath.org/authors/?q=ai:zhang.yiyingSummary: In reliability engineering, the relevation model can be adopted to characterize the performance of redundancy allocation for coherent systems. In this paper, we investigate the allocation problems of relevations for two nodes in a coherent system with independent components for enhancing system reliability. We first investigate the optimal allocation policy of two relevations for two nodes of the system under certain conditions. As a special setting of the relevation, we further discuss optimal allocation strategies for a batch of minimal repairs allocated to two components of the coherent system by applying the useful tool of majorization order. Sufficient conditions are established in terms of structural relationships between the components induced by minimal cut or path sets and the reliabilities of components and relevations. Some numerical examples are provided as illustrations. A real application in aircraft indicator lights systems is also presented to show the availability of our results.Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of \(L^p (\mathbb{R}^n)\)https://zbmath.org/1528.940172024-03-13T18:33:02.981707Z"Patel, Dhiraj"https://zbmath.org/authors/?q=ai:patel.dhiraj"Sivananthan, S."https://zbmath.org/authors/?q=ai:sivananthan.sivalingamSummary: The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on \(\Omega\) (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on \(\Omega\). Moreover, we prove with an overwhelming probability that \(\mathcal{O}(\mu (\Omega )(\log \mu (\Omega ))^3)\) many random points uniformly distributed over \(\Omega\) yield a stable set of sampling for functions concentrated on \(\Omega\).