The Exact Worst-Case Tail Probability under Bounded Kurtosis

Abstract

We determine exactly what a kurtosis bound buys for one-sided tail control. For the class C(κ) of real random variables with mean 0, variance 1, and fourth moment at most κ, the skewness left free (the form in which fourth-moment information typically arrives in learning theory and robust statistics), we compute the worst-case tail probability V₁(t, κ) = sup_{X∈C(κ)} P(X ≥ t) for every threshold t > 0 and every κ ≥ 1. The answer is a four-regime map: a Cantelli tongue b(κ) ≤ t ≤ c(κ) on which the two-moment bound 1/(1 + t²) remains tight and the kurtosis constraint is worthless; a tail regime t ≥ c(κ) with the closed form V₁ = (κ − 1)/((t² − 1)² + κ − 1); a plateau regime, present only for κ ≤ 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√3 − 3 constant of He et al. (2010) at t = 0. Code, certificates, and the full instance tables are available at https://github.com/xiaoyulics/lemmaforge.