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Register a new userA simplified second-order Gaussian Poincare inequality in discrete setting with applications
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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.
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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 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.
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 $xinmathbb{R}^d$, $sigma:mathbb{R}tomathbb{R}$ 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 function $f$. 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 of such discussions. The proof rests on novel facts about functions of positive type, and on strong localization bounds for comparison of SPDEs.
We improve the constant $frac{pi}{2}$ in $L^1$-Poincare inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $sqrt{frac{pi}{2}}$. For Hamming cube the sharp constant is not known, and $sqrt{frac{pi}{2}}$ gives an estimate from below for this sharp constant. On the other hand, L. Ben Efraim and F. Lust-Piquard have shown an estimate from above: $C_1le frac{pi}{2}$. There are at least two other independent proofs of the same estimate from above (we write down them in this note). Since those proofs are very different from the proof of Ben Efraim and Lust-Piquard but gave the same constant, that might have indicated that constant is sharp. But here we give a better estimate from above, showing that $C_1$ is strictly smaller than $frac{pi}{2}$. It is still not clear whether $C_1> sqrt{frac{pi}{2}}$. We discuss this circle of questions and the computer experiments.
In this paper we present a new formulation of the change of gauge formulas in second order cosmological perturbation theory which unifies and simplifies known results. Our approach is based on defining new second order scalar perturbation variables by adding a multiple of the square of the corresponding first order variables to each second order variable. A bonus is that these new perturbation variables are of broader significance in that they also simplify the analysis of second order scalar perturbations in the super-horizon regime in a number of ways, and lead to new conserved quantities.
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