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Register a new userSecond order Poincare inequalities and CLTs on Wiener space
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We prove infinite-dimensional second order Poincare inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Steins method and Malliavin calculus. We provide two applications: (i) to a new second order characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.
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Given a vector $F=(F_1,dots,F_m)$ of Poisson functionals $F_1,dots,F_m$, we investigate the proximity between $F$ and an $m$-dimensional centered Gaussian random vector $N_Sigma$ with covariance matrix $Sigmainmathbb{R}^{mtimes m}$. Apart from finding proximity bounds for the $d_2$- and $d_3$-distances, based on classes of smooth test functions, we obtain proximity bounds for the $d_{convex}$-distance, based on the less tractable test functions comprised of indicators of convex sets. The bounds for all three distances are shown to be of the same order, which is presumably optimal. The bounds are multivariate counterparts of the univariate second order Poincare inequalities and, as such, are expressed in terms of integrated moments of first and second order difference operators. The derived second order Poincare inequalities for indicators of convex sets are made possible by a new bound on the second derivatives of the solution to the Stein equation for the multivariate normal distribution. We present applications to the multivariate normal approximation of first order Poisson integrals and of statistics of Boolean models.
We prove a family of Hardy-Rellich and Poincare identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincare inequalities. All remainder terms provided considerably improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second ord
In this paper, a simplified second-order Gaussian Poincare 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 Erdos-Renyi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random $kappa$-complex and infinite weighted 2-runs are treated.
Consider a parabolic stochastic PDE of the form $partial_t u=frac{1}{2}Delta u + sigma(u)eta$, where $u=u(t,,x)$ for $tge0$ and $xinmathbb{R}^d$, $sigma:mathbb{R}rightarrowmathbb{R}$ is Lipschitz continuous and non random, and $eta$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation $f$. If, in addition, $u(0)equiv1$, then we prove that, under a mild decay condition on $f$, the process $xmapsto u(t,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of such discussions. Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincare inequalities. We further showcase the utility of these Poincare inequalities by: (a) describing conditions that ensure that the random field $u(t)$ is mixing for every $t>0$; and by (b) giving a quick proof of a conjecture of Conus et al cite{CJK12} about the size of the intermittency islands of $u$. The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama cite{Maruyama} (see also Dym and McKean cite{DymMcKean}) in the simple setting where the nonlinear term $sigma$ is a constant function.
In 1991, Persi Diaconis and Daniel Stroock obtained two canonical path bounds on the second largest eigenvalue for simple random walk on a connected graph, the Poincare and Cheeger bounds, and they raised the question as to whether the Poincare bound is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincare, establish some sufficient conditions on the canonical paths for the Poincare bound to triumph, and show that there is always a choice of paths for which this happens.
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