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Register a new userGaussian limit for determinantal point processes with $J$-Hermitian kernels
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We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.
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The gamma kernels are a family of projection kernels $K^{(z,z)}=K^{(z,z)}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Eulers gamma function and depend on two continuous parameters $z,z$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z)}$ serves as a correlation kernel for a determinantal measure $M^{(z,z)}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ ldots, K^{(z-1,z-1)}, ; K^{(z,z)},; K^{(z+1,z+1)}, ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z)$, the kernel $K^{(z,z)}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z+1)}$, and the one-point Palm distributions for the measure $M^{(z,z)}$ are absolutely continuous with respect to $M^{(z+1,z+1)}$. We also explicitly compute the corresponding Radon-Nikodym derivatives and show that they are given by certain normalized multiplicative functionals.
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, omega)$ over a domain $U subset mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^infty(U)$-module structure of $A^2(U, omega)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.
We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| to infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for ${E(k;J)}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.
Let $Y_i,igeq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $mathcal {M}subset mathbb{R}^d$ and consider sums $sum_{i=1}^nxi(n^{1/m}Y_i,{n^{1/m}Y_j}_{j=1}^n)$, where $xi$ is a real valued function defined on pairs $(y,mathcal {Y})$, with $yin mathbb{R}^d$ and $mathcal {Y}subset mathbb{R}^d$ locally finite. Subject to $xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $mathcal {M}$. We apply the general results to establish the limit theory of dimension and volume content estimators, R{e}nyi and Shannon entropy estimators and clique counts in the Vietoris-Rips complex on ${Y_i}_{i=1}^n$.
Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution $q$ over the subsets of a ground set, we aim to distinguish whether $q$ is a DPP distribution, or $epsilon$-far from all DPP distributions in $ell_1$-distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of log-submodular distributions.
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