Related papers: A Counterexample to the Vector Generalization of C…

In a recent paper `The equi-energy sampler with applications statistical inference and statistical mechanics' [Ann. Stat. 34 (2006) 1581--1619], Kou, Zhou & Wong have presented a new stochastic simulation method called the equi-energy (EE)…

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in…

Recently we proved (EPRSY, ERSTVY, ESY4, ESYY, EYY, EYY2, EYYrigi) that the eigenvalue correlation functions of a general class of random matrices converge, weakly with respect to the energy, to the corresponding ones of Gaussian matrices.…

We show that the convolution of a compactly supported measure on $\mathbb{R}$ with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). We use this result to give a new proof of a classical result in random matrix theory…

We prove noncommutative martingale inequalities associated with convex functions. More precisely, we obtain $\Phi$-moment analogues of the noncommutative Burkholder inequalities and the noncommutative Rosenthal inequalities for any convex…

We show that a problem by Yau can not be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng.

In this paper, we generalize some matrix inequalities involving matrix power and Karcher means of positive definite matrices. Among other inequalities, it is shown that if ${\mathbb A}=(A_{1},...,A_{n})$ is a $n$-tuple of positive definite…

For statistical systems that violate one of the four Shannon-Khinchin axioms, entropy takes a more general form than the Boltzmann-Gibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with…

Given two density matrices $\rho$ and $\sigma$, there are a number of different expressions that reduce to the $\alpha$-R\'enyi relative entropy of $\rho$ with respect to $\sigma$ in the classical case; i.e., when $\rho$ and $\sigma$…

We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic…

We use Matrix Analysis to prove a general decoupling inequality for finite Gaussian vectors, in identifying a new region of the inherent $p$ exponent, for the validity of this one.

We give a necessary and sufficient condition for the incompatibility of a pair of effects in a general probabilistic model in which the state space is a total convex space, which can be obtained by minimising a real parameter. This has an…

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\|\mathbf{v}^2\|\|\mathbf{w}^2\| - \langle\mathbf{v}^2,\mathbf{w}^2\rangle \leq…

We give an alternative proof to Wu's logarithmic Sobolev inequality for the Poisson measure on the nonnegative integers using a stochastic variational formula for entropy. We show that this approach leads to improvement of Wu's inequality…

The contraction inequality for Rademacher averages is extended to Lipschitz functions with vector-valued domains, and it is also shown that in the bounding expression the Rademacher variables can be replaced by arbitrary iid symmetric and…

We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive L\'evy noise. We first derive a representation formula for the error which we then…

The adjoint method introduced in [Eva] and [Tra] is used, to construct analogs to the Aubry-Mather measures for non convex Hamiltonians. More precisely, a general construction of probability measures, that in the convex setting agree with…

The joint convexity of the map $(X,A) \mapsto X^* A^{-1} X$, an integral representation of operator convex functions, and an observation of Ando are used to obtain a simple proof of both the joint convexity of relative entropy and a trace…

Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,...,a_{g_a},x_1,...,x_{g_x})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $p = L + \Lambda ^T \Lambda,$ where $L$…

In this paper, we study some properties such as the monotonicity, logarithmically complete monotonicity, logarithmic convexity, and geometric convexity, of the combinations of gamma function and power function. The results we obtain…