Variance bounds in product measures without exponential tails
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914272103104512 |
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| author | Feng, Shi |
| author_facet | Feng, Shi |
| contents | 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 $λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_15450 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variance bounds in product measures without exponential tails Feng, Shi Probability Functional Analysis 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 $λ$. |
| title | Variance bounds in product measures without exponential tails |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2601.15450 |