The Exact Worst-Case Tail Probability under Bounded Kurtosis

2026-07-06

AI Summary

This work provides the first complete characterization of worst-case one-sided tail probabilities under the sole assumptions that a random variable has zero mean, unit variance, and fourth moment bounded by κ. By integrating semidefinite programming duality, sum-of-squares (SOS) proof systems, symbolic computation, and an AI-guided LemmaForge reasoning framework, the authors exactly solve for the maximal tail probability at any threshold t, revealing four distinct phase-transition regimes as t varies. Closed-form expressions or algebraic characterizations are derived for each regime. The study demonstrates the failure of Cantelli’s bound in the high-tail region, constructs extremal distributions along with dual certificates, derives exact quantiles, improves the constant in median-of-means estimation, and unifies and recovers several classical results within a single coherent framework.

Abstract

We determine exactly what a kurtosis bound buys for one-sided tail control. For the class $\mathcal{C}(κ)$ of real random variables with mean $0$, variance $1$, and fourth moment at most $κ$, the skewness left free, we compute the worst-case tail probability $V_1(t,κ)=\sup_{X\in\mathcal{C}(κ)}\mathbb{P}(X\geq t)$ for every threshold $t>0$ and every $κ\geq 1$. The answer is a four-regime map: a Cantelli tongue $b(κ)\le t\le c(κ)$ on which the two-moment bound $1/(1+t^2)$ remains tight and the kurtosis constraint is worthless; a tail regime $t\geq c(κ)$ with the closed form $V_1=(κ-1)/((t^2-1)^2+κ-1)$; a plateau regime, present only for $κ\le 3/2$, on which the worst case freezes and the value does not depend on $t$; and a central regime described exactly by an explicit algebraic system, provably admitting no closed form in nested square roots. Beyond $c(κ)$ the one-sided and two-sided worst cases coincide: Cantelli's improvement over Chebyshev is annihilated by fourth-moment information. The minimal degree of a sum-of-squares proof of the tight bound is $2$ on the closed tongue and $4$ everywhere else, an exact phase diagram of proof degree. Every closed-form regime carries an explicit dual certificate and an explicit extremal distribution, re-verified on parameter grids by an independent checker in exact arithmetic. The closed forms invert to exact worst-case quantiles, sharpen a median-of-means constant, and give the exact per-direction tail available to degree-4 reasoning under certifiable kurtosis. We found the map through an AI-guided search around the certifying pipeline, LemmaForge, which is validated on classical benchmarks, independently reproduces the symmetric-slice bound of Zelen (1954), and recovers the $2\sqrt{3}-3$ constant of He, Zhang, and Zhang (2010) at $t=0$.

Problem

• kurtosis
• tail probability
• worst-case analysis
• moment constraints
• one-sided tail

Innovation

• exact tail bounds
• bounded kurtosis
• sum-of-squares proof
• extremal distributions
• AI-guided theorem discovery