Multivariate CLT follows from strong Rayleigh property

Let $(X_1 , \ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, \ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~\cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.

[1]  Edward A. Bender,et al.  Central and Local Limit Theorems Applied to Asymptotic Enumeration , 1973, J. Comb. Theory A.

[2]  J. Borcea,et al.  The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability , 2008, 0809.0401.

[3]  Yuval Peres,et al.  Concentration of Lipschitz Functionals of Determinantal and Other Strong Rayleigh Measures , 2011, Combinatorics, Probability and Computing.

[4]  Alexander Soshnikov.  Gaussian limit for determinantal random point fields , 2000 .

[5]  Edward A. Bender,et al.  Central and Local Limit Theorems Applied to Asymptotic Enumeration II: Multivariate Generating Functions , 1983, J. Comb. Theory, Ser. A.

[6]  T. Liggett,et al.  Negative dependence and the geometry of polynomials , 2007, 0707.2340.

[7]  A. Soshnikov.  Determinantal random point fields , 2000, math/0002099.

[8]  Sean X. Chen,et al.  General Properties and Estimation of Conditional Bernoulli Models , 2000 .

[9]  Boris G. Pittel,et al.  Central limit theorems, Lee-Yang zeros, and graph-counting polynomials , 2014, J. Comb. Theory, Ser. A.

[10]  N. Butt.  Sampling with Unequal Probabilities , 2003 .

[11]  Julius Borcea,et al.  The Lee‐Yang and Pólya‐Schur programs. II. Theory of stable polynomials and applications , 2008, 0809.3087.

[12]  Thomas Ransford,et al.  When is a probability measure determined by infinitely many projections , 1997 .

[13]  David G. Wagner,et al.  Homogeneous multivariate polynomials with the half-plane property , 2004, Adv. Appl. Math..

[14]  Mark C. Wilson,et al.  Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions , 2005, SIAM Rev..

[15]  Mark C. Wilson,et al.  Asymptotics of Multivariate Sequences II: Multiple Points of the Singular Variety , 2004, Combinatorics, Probability and Computing.

[16]  Normalized symmetric functions, Newton's inequalities, and a new set of stronger inequalities , 1989 .

[17]  Robin Pemantle,et al.  Hyperbolicity and stable polynomials in combinatorics and probability , 2012, 1210.3231.

[18]  Torin Greenwood.  ASYMPTOTICS OF THE COEFFICIENTS OF BIVARIATE ANALYTIC FUNCTIONS WITH ALGEBRAIC SINGULARITIES (PREPRINT) , 2014 .

[19]  Mohit Singh,et al.  A Randomized Rounding Approach to the Traveling Salesman Problem , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[20]  Johan Jonasson,et al.  Negative Dependence in Sampling , 2012 .