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Variance bounds in product measures without exponential tails

Shi Feng

TL;DR

The paper develops α-Cheeger and α-Poincaré-type inequalities to obtain dimension-dependent variance bounds for 1-Lipschitz functions under product measures with heavy tails, including Pareto distributions. By linking tail information to quantile behavior through a quantile-based framework and key truncation lemmas, it derives sharp $L^1$ and $L^2$ moment bounds and then tensorizes them to high dimensions, yielding $Var_{\mu^n}(f) = O\left(n^{2/(\lambda-1)}\right)$ for Pareto measures with $\lambda>3$. The results extend to general metrics $d_p$, provide isoperimetric and fluctuation consequences, and have applications to the variance of eigenvalues in heavy-tailed random matrices; the bounds are shown to be asymptotically optimal in the Lipschitz setting. Overall, the work broadens functional-inequality methods to heavy-tailed product measures and yields explicit, dimension-sensitive variance control for 1-Lipschitz observables.

Abstract

We establish analogs of Cheeger's inequality for probability measures with heavy tails. As one of the principal applications, suppose $λ> 3$ and define the (Pareto) probability measure $μ_λ$ on $[1,\infty)$ by $dμ_λ(x) = (λ- 1) x^{-λ}$. Let $μ_λ^n$ denote the product measure of $μ_λ$ on $\mathbb{R}^n$. Then, for any $1$-Lipschitz function (with respect to the Euclidean distance) $f : \mathbb{R}^n \to \mathbb{R}$, we obtain the variance bound $\operatorname{Var}_{μ_λ^n}(f) \le C(λ)\, n^{\frac{2}{λ- 1}}$, where $C(λ)$ is an explicit constant depending only on $λ$. This improves upon the existing bound $\operatorname{Var}_{μ_λ^n}(f) = O(n)$ derived from the Efron--Stein inequality. Moreover, this bound is asymptotically tight when considering the $1$-Lipschitz function $f(x) = |x|_{\infty}$ corresponding to the $L^{\infty}$ norm. In probabilistic terms, suppose $X_1, \dots, X_n$ are i.i.d.\ random variables with distribution $μ_λ$. Then, for any $1$-Lipschitz function $f$, we have $\operatorname{Var}(f(X_1, \dots, X_n)) \le C'(λ)\operatorname{Var}(\max\{X_1, \dots, X_n\}) = Θ\!\left(n^{\frac{2}{λ- 1}}\right)$, where $C'(λ)$ is another explicit constant depending only on $λ$.

Variance bounds in product measures without exponential tails

TL;DR

The paper develops α-Cheeger and α-Poincaré-type inequalities to obtain dimension-dependent variance bounds for 1-Lipschitz functions under product measures with heavy tails, including Pareto distributions. By linking tail information to quantile behavior through a quantile-based framework and key truncation lemmas, it derives sharp and moment bounds and then tensorizes them to high dimensions, yielding for Pareto measures with . The results extend to general metrics , provide isoperimetric and fluctuation consequences, and have applications to the variance of eigenvalues in heavy-tailed random matrices; the bounds are shown to be asymptotically optimal in the Lipschitz setting. Overall, the work broadens functional-inequality methods to heavy-tailed product measures and yields explicit, dimension-sensitive variance control for 1-Lipschitz observables.

Abstract

We establish analogs of Cheeger's inequality for probability measures with heavy tails. As one of the principal applications, suppose and define the (Pareto) probability measure on by . Let denote the product measure of on . Then, for any -Lipschitz function (with respect to the Euclidean distance) , we obtain the variance bound , where is an explicit constant depending only on . This improves upon the existing bound derived from the Efron--Stein inequality. Moreover, this bound is asymptotically tight when considering the -Lipschitz function corresponding to the norm. In probabilistic terms, suppose are i.i.d.\ random variables with distribution . Then, for any -Lipschitz function , we have , where is another explicit constant depending only on .
Paper Structure (16 sections, 22 theorems, 191 equations)

This paper contains 16 sections, 22 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Technical lemmas
  3. Proof of Theorems \ref{['L^1 theorem']}, \ref{['L^2 theorem']}, and \ref{['L^1 Cheeger theorem']}
  4. Proof of Theorem \ref{['L^1 theorem']}
  5. Proof of Theorem \ref{['L^2 theorem']}
  6. Proof of Theorem \ref{['L^1 Cheeger theorem']}
  7. Proof of Theorems \ref{['product theorem']} and Theorem \ref{['Pareto theorem']}
  8. Proof of Theorem \ref{['product theorem']}
  9. Proof of Theorem \ref{['Pareto theorem']}
  10. Proof of Theorem \ref{['d_p theorem']}
  11. Consequences and extensions
  12. Fluctuation bound for 1-Lipschitz functions by Cheeger/$\alpha$-Cheeger constants
  13. Isoperimetric inequalities for product measures
  14. 1-Lipschitz function with respect to $d_p (1<p\leq2)$ by traditional Poincaré constant
  15. Applications to variance of eigenvalues for heavy tail random matrices
  16. ...and 1 more sections

Key Result

Proposition 1.1

Suppose $\nu$ is a non-degenerate $\kappa$-concave probability measure on $\mathbb{R}^n$ for some $-\infty<\kappa\leq 0$. Denote $m := m_{\nu}(|x|)$. Then where $c(\kappa)$ is a positive continuous function on $(-\infty,0]$.

Theorems & Definitions (43)

  • Definition 1.1
  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.1
  • ...and 33 more