Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures

Details Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures

2011-08-02

arxiv.org/abs/1108.0687v1

Mathematics

Probability

Scientific paper

Let X_1 ,..., X_n be a collection of binary valued random variables and let f : {0,1}^n -> R be a Lipschitz function. Under a negative dependence hypothesis known as the {\em strong Rayleigh} condition, we show that f - E f satisfies a concentration inequality generalizing the classical Gaussian concentration inequality for sums of independent Bernoullis: P (S_n - E S_n > a) < exp (-2 a^2 / n). The class of strong Rayleigh measures includes determinantal measures, weighted uniform matroids and exclusion measures; some familiar examples from these classes are generalized negative binomials and spanning tree measures. For instance, the number of vertices of odd degree in a uniform random spanning tree of a graph satisfies a Gaussian concentration inequality with n replaced by |V|, the number of vertices. We also prove a continuous version for concentration of Lipschitz functionals of a determinantal point process.

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