Variance bounds in product measures without exponential tails

Abstract

We establish analogs of Cheeger's inequality for probability measures with heavy tails. As one of the principal applications, suppose $\lambda > 3$ and define the (Pareto) probability measure $\mu_{\lambda}$ on $[1,\infty)$ by $d\mu_{\lambda}(x) = (\lambda - 1) x^{-\lambda}$. Let $\mu_{\lambda}^n$ denote the product measure of $\mu_{\lambda}$ 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 $\operatorname{Var}_{\mu_{\lambda}^n}(f) \le C(\lambda)\, n^{\frac{2}{\lambda - 1}}$, where $C(\lambda)$ is an explicit constant depending only on $\lambda$. This improves upon the existing bound $\operatorname{Var}_{\mu_{\lambda}^n}(f) = O(n)$ 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 $\mu_{\lambda}$. Then, for any $1$-Lipschitz function $f$, we have $\operatorname{Var}(f(X_1, \dots, X_n)) \le C'(\lambda)\operatorname{Var}(\max\{X_1, \dots, X_n\}) = \Theta\!\left(n^{\frac{2}{\lambda - 1}}\right)$, where $C'(\lambda)$ is another explicit constant depending only on $\lambda$.