[Paper Review] Weighted Poincar\'e inequalities, concentration inequalities and tail bounds related to the behavior of the Stein kernel in dimension one

Adrien Saumard|arXiv (Cornell University)|Apr 11, 2018

Random Matrices and Applications35 references7 citations

TL;DR

This paper establishes connections between Stein's density approach in one dimension and functional inequalities, proving weighted Poincaré and Brascamp-Lieb inequalities using the Stein kernel as a weight. It derives new concentration and tail bounds, including generalized Mills' inequalities and sub-Gamma concentration for Lipschitz functions, with a key contribution being a general lemma for bounding Laplace transforms of random variables to enable concentration results.

ABSTRACT

We investigate the links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a weighted Poincar\'e inequality with the weight being the Stein kernel. Furthermore we prove asymmetric Brascamp-Lieb type inequalities related to the Stein kernel. We also show that existence of a uniformly bounded Stein kernel is sufficient to ensure a positive Cheeger isoperimetric constant. Then we derive new concentration inequalities. In particular, we prove generalized Mills' type inequalities when the Stein kernel is uniformly bounded and sub-gamma concentration for Lipschitz functions of a variable with sub-linear Stein kernel. When some exponential moments are finite, a general concentration inequality is then expressed in terms of Legendre-Fenchel transform of the Laplace transform of the Stein kernel. Along the way, we prove a general lemma for bounding the Laplace transform of a random variable, that should be very useful in many other contexts when deriving concentration inequalities. Finally, we provide density and tail formulas as well as tail bounds, generalizing previous results that where obtained in the context of Malliavin calculus.

Motivation & Objective

• To explore the relationship between the Stein kernel and functional inequalities in one-dimensional probability distributions.
• To establish weighted Poincaré and Brascamp-Lieb type inequalities using the Stein kernel as a weight.
• To derive new concentration and tail bounds for random variables with sub-linear or uniformly bounded Stein kernels.
• To develop a general lemma for bounding Laplace transforms of random variables to facilitate concentration inequality derivation.
• To generalize existing results on density and tail formulas beyond the Malliavin calculus framework.

Proposed method

• Derives weighted Poincaré inequalities by using the Stein kernel as the weight function for measures with finite first moment and connected support.
• Proves asymmetric Brascamp-Lieb inequalities based on the structure of the Stein kernel in one dimension.
• Establishes a positive Cheeger isoperimetric constant under the condition of uniformly bounded Stein kernel.
• Applies a general lemma for bounding the Laplace transform of a random variable to derive concentration inequalities.
• Expresses concentration inequalities in terms of the Legendre-Fenchel transform of the Laplace transform of the Stein kernel when exponential moments are finite.
• Derives explicit density and tail formulas, generalizing prior results from Malliavin calculus to broader classes of distributions.

Experimental results

Research questions

• RQ1 How does the Stein kernel relate to weighted Poincaré inequalities in one-dimensional distributions?
• RQ2 Can asymmetric Brascamp-Lieb inequalities be derived using the Stein kernel as a structural component?
• RQ3 Under what conditions on the Stein kernel does a positive Cheeger constant emerge?
• RQ4 What concentration inequalities can be derived when the Stein kernel is uniformly bounded or sub-linear?
• RQ5 How can the Laplace transform of a random variable be bounded to enable general concentration results?

Key findings

• Measures with finite first moment and connected support satisfy a weighted Poincaré inequality with the Stein kernel as the weight function.
• Asymmetric Brascamp-Lieb type inequalities are established, linking the Stein kernel to functional inequalities in one dimension.
• A uniformly bounded Stein kernel ensures a positive Cheeger isoperimetric constant, implying strong concentration properties.
• Generalized Mills' type inequalities are derived when the Stein kernel is uniformly bounded.
• Sub-Gamma concentration is established for Lipschitz functions of random variables with sub-linear Stein kernels.
• A general concentration inequality is expressed via the Legendre-Fenchel transform of the Laplace transform of the Stein kernel when exponential moments are finite.