Abstract
We study the weighted Poincaré constant of a probability density p with weight function w using integration methods inspired by Stein’s method. We obtain a new version of the Chen-Wang variational formula which, as a byproduct, yields simple upper and lower bounds on in terms of the so-called Stein kernel of p. We also iterate these variational formulas so as to build sequences of nested intervals containing the Poincaré constant, sequences of functions converging to said constant, as well as sequences of functions converging to the solutions of the corresponding spectral problem. Our results rely on the properties of a pseudo inverse operator of the classical Sturm-Liouville operator. We illustrate our methods on a variety of examples: Gaussian functionals, weighted Gaussian, beta, gamma, Subbotin, and Weibull distributions.
Acknowledgements
GG acknowledges support of the grant “Action de Recherche Concertées –projets consolidation 2020-2023” from ULB. YS acknowledges partial support from grant CDR/OL J.0197.20 from FRS-FNRS. The authors thank the AE and referee for their comments on the previous version of the paper.
Citation
Gilles Germain. Yvik Swan. "A note on one-dimensional Poincaré inequalities by Stein-type integration." Bernoulli 29 (2) 1714 - 1740, May 2023. https://doi.org/10.3150/22-BEJ1518
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