New representations of Stein operators produce explicit weighted covariance identities that deliver sharp upper and lower covariance bounds and weighted Poincaré inequalities for univariate targets, recovering classical results as corollaries.
On the isoperimetric constant, covariance inequalities and $L_p$-Poincar\'{e} inequalities in dimension one
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abstract
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger's inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincar\'{e} inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincar\'{e} inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger's inequality, which is a $L_{p}$-Poincar\'{e} inequality for $p=2$, to any real $p\geq 1$.
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math.PR 1years
2019 1verdicts
UNVERDICTED 1representative citing papers
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First order covariance inequalities via Stein's method
New representations of Stein operators produce explicit weighted covariance identities that deliver sharp upper and lower covariance bounds and weighted Poincaré inequalities for univariate targets, recovering classical results as corollaries.