We consider probability measures supported on a finite discrete interval $[0,n]$. We introduce a new finitedifference operator $\nabla_n$, defined as a linear combination of left and right finite differences. We show that this operator $\nabla_n$ plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures.
We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian $d\times d$ matrix valued functions, with the same Poincaré constant. This generalizes recent results [Aoun et al. 2019, Kathuria 2019] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.
Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纪念大学格伦菲分校科学与环境学院）；Department of Mathematics, University of Connecticut（康涅狄格大学数学系）
An Improved Second Order Poincaré Inequality for Functionals of Gaussian Fields
本文给出了Chatterjee（2009）和Nourdin，Peccati and Reinert（2009）首次提出的二阶Gaussian Poincaré不等式的一个改进形式。这些新的估计是用来约束分布距离之间的泛函的高斯场和正常的随机变量。几个应用程序的开发，包括定量CLTs的非线性泛函的平稳高斯场相关的布鲁尔-主要定理，改善以前的研究结果在文献中，并获得假定最佳的收敛速度。
This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument.
This paper is concerned with the problem of nonlinear filter stability of ergodic Markov processes. The main contribution is the conditional Poincaré inequality (PI), which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE). Based on these dual formalisms, a comparison is drawn between the stochastic stability of a Markov process and the filter stability. The latter relies on the conditional PI describ…
Inspired by the idea of stochastic quantization proposed by Parisi and Wu, we construct the transition probability matrix which plays a central role in the renormalization group through a stochastic differential equation. By establishing the discrete time stochastic dynamics, the renormalization procedure can be characterized from the perspective of probability. Hence, we will focus on the investigation of the infinite dimensional stochastic dynamic. From the stochastic point of view, the discrete time stochastic dynamic can induce a Markov chain. Via calculating the square field operator and …
In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdös-Rényi random graph are discussed. The number of vertices of fixed degree is also studie…
We study a sparse negative binomial regression (NBR) for count data by showing the non-asymptotic advantages of using the elastic-net estimator. Two types of oracle inequalities are derived for the NBR's elastic-net estimates by using the Compatibility Factor Condition and the Stabil Condition. The second type of oracle inequality is for the random design and can be extended to many $\ell_1 + \ell_2$ regularized M-estimations, with the corresponding empirical process having stochastic Lipschitz properties. We derive the concentration inequality for the suprema empirical processes for the weigh…
We prove the inequality $E[(X/μ)^k] \le (\frac{k/μ}{\log(k/μ+1)})^k \le \exp(k^2/(2μ))$ for sub-Poissonian random variables, such as Binomially or Poisson distributed random variables with mean $μ$. The asymptotics $1+O(k^2/μ)$ can be shown to be tight for small $k$. This improves over previous uniform bounds for the raw moments of those distributions by a factor exponential in $k$.
