Recent zbMATH articles in MSC 60E15https://zbmath.org/atom/cc/60E152024-03-13T18:33:02.981707ZWerkzeugSharp 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.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.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 \(\Omega\).