Explicit Universal Bounds for Cumulants via Moments
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910129338712064 |
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| author | Zhang, Jiechen |
| author_facet | Zhang, Jiechen |
| contents | We establish explicit, universal, and distribution-free bounds for the $n$-th cumulant, $κ_n(X)$, of a scalar random variable, controlled solely by an $n$-th order absolute moment functional $M_n(X)$. The bounds take the form $\lvertκ_n(X)\rvert \le C_n M_n(X)$. Our principal contribution is the derivation of coefficients satisfying $C_n \sim (n-1)!/ρ^{\,n}$, which offers an exponential improvement over classical bounds where the coefficients grow superexponentially (on the order of $n^n$).
We present a hierarchy of refinements where the rate parameter $ρ$ increases as the functional $M_n(X)$ incorporates more structural information. The most general bound uses the raw moment $M_n(X)=\mathsf{E}[\lvert X\rvert^n]$ with rate $ρ=\ln 2 \approx 0.693$. Using the central moment $M_n(X)=\mathsf{E}[\lvert X-\mathsf{E}[X]\rvert^n]$ improves the rate to $ρ_{\mathrm{cen}} \approx 1.146$, while assuming symmetry yields even higher rates.
The proof is elementary, combining the moment-cumulant partition formula with a uniform moment-product inequality. We further prove that while these bounds are not attainable whenever the relevant coefficient is positive, they are asymptotically efficient given the limited information of a single moment. The utility of the bounds is demonstrated through an application to standardized cumulants of independent sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Universal Bounds for Cumulants via Moments Zhang, Jiechen Probability Combinatorics Statistics Theory We establish explicit, universal, and distribution-free bounds for the $n$-th cumulant, $κ_n(X)$, of a scalar random variable, controlled solely by an $n$-th order absolute moment functional $M_n(X)$. The bounds take the form $\lvertκ_n(X)\rvert \le C_n M_n(X)$. Our principal contribution is the derivation of coefficients satisfying $C_n \sim (n-1)!/ρ^{\,n}$, which offers an exponential improvement over classical bounds where the coefficients grow superexponentially (on the order of $n^n$). We present a hierarchy of refinements where the rate parameter $ρ$ increases as the functional $M_n(X)$ incorporates more structural information. The most general bound uses the raw moment $M_n(X)=\mathsf{E}[\lvert X\rvert^n]$ with rate $ρ=\ln 2 \approx 0.693$. Using the central moment $M_n(X)=\mathsf{E}[\lvert X-\mathsf{E}[X]\rvert^n]$ improves the rate to $ρ_{\mathrm{cen}} \approx 1.146$, while assuming symmetry yields even higher rates. The proof is elementary, combining the moment-cumulant partition formula with a uniform moment-product inequality. We further prove that while these bounds are not attainable whenever the relevant coefficient is positive, they are asymptotically efficient given the limited information of a single moment. The utility of the bounds is demonstrated through an application to standardized cumulants of independent sums. |
| title | Explicit Universal Bounds for Cumulants via Moments |
| topic | Probability Combinatorics Statistics Theory |
| url | https://arxiv.org/abs/2510.05739 |