Bounds on the Poincaré constant for convolution measures

2018/06/29 by Courtade, Thomas A. · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Information Theory (cs.IT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1807.00027

Abstract

We establish a Shearer-type inequality for the Poincaré constant, showing that the Poincaré constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincaré constants corresponding to convolutions of subsets of measures. This implies, for example, that the Poincaré constant is non-increasing along the central limit theorem. We also establish a dimension-free stability estimate for subadditivity of the Poincaré constant on convolutions which uniformly improves an earlier one-dimensional estimate of a similar nature by Johnson (2004). As a byproduct of our arguments, we find that the monotone properties of entropy, Fisher information and the Poincaré constant along the CLT find a common root in Shearer's inequality.

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