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 $λ&gt; 3$ and define the (Pareto) probability measure $μ<em>λ$ on $[1,\infty)$ by $dμ</em>λ(x) = (λ- 1) x<sup>{-λ}$. Let $μ<em>λ<sup>n$ denote the product measure of $μ</em>λ$ on $\mathbb{R}<sup>n$. Then, for any $1$-Lipschitz function (with respect to the Euclidean distance) $f : \mathbb{R}ⁿ \to \mathbb{R}$, we obtain the variance bound $$\operatorname{Var}<em>{μ</em>λ^n}(f) \le C(λ)\, n^{{\frac{2}{λ-} 1}}$$, where $C(λ)$ is an explicit constant depending only on $λ$. This improves upon the existing bound $$\operatorname{Var}<em>{μ</em>λ^n}(f) = O(n)$$ derived from the Efron--Stein inequality. Moreover, this bound is asymptotically tight when considering the $1$-Lipschitz function $f(x) = |x|<em>{\infty}$ corresponding to the $L<sup>{\infty}$ norm. In probabilistic terms, suppose $X_1, \dots, X_n$ are i.i.d.\ random variables with distribution $μ</em>λ$. Then, for any $1$-Lipschitz function $f$, we have $\operatorname{Var}(f(X_1, \dots, X_n)) \le C&#39;(λ)\operatorname{Var}(\max{X_1, \dots, X_n}) = Θ!\left(n^{{\frac{2}{λ-} 1}}\right)$, where $C&#39;(λ)$ is another explicit constant depending only on $λ$.