Variance bounds in product measures without exponential tails

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 $[1,\mathrm{\infty })$ 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 ${\mathrm{Var}}_{{\mu }_{\lambda }^n}(f)\le C(\lambda )n^{\frac{2}{\lambda -1}}$, where $C(\lambda )$ is an explicit constant depending only on $\lambda$. This improves upon the existing bound ${\mathrm{Var}}_{{\mu }_{\lambda }^n}(f)=O(n)$ derived from the Efron–Stein inequality. Moreover, this bound is asymptotically tight when considering the $1$-Lipschitz function $f(x)=\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}$ corresponding to the $L^{\mathrm{\infty }}$ 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 $\mathrm{Var}(f(X_1,\dots ,X_n))\le C^{\mathrm{\prime }}(\lambda )\mathrm{Var}(\max \{X_1,\dots ,X_n\})=\mathrm{\Theta }\text{\hspace{0.17em}⁣}\left(n^{\frac{2}{\lambda -1}}\right)$, where $C^{\mathrm{\prime }}(\lambda )$ is another explicit constant depending only on $\lambda$.