# The Exact Worst-Case Tail Probability under Bounded Kurtosis

arXiv: https://arxiv.org/abs/2607.05226 · HTML: https://mlsanity.com/paper/2607.05226/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 32 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Conformer: Convolution-augmented Transformer for Speech Recognition](https://mlsanity.com/paper/2005.08100/) (2020)
- [Subword Regularization: Improving Neural Network Translation Models with Multiple Subword Candidates](https://mlsanity.com/paper/1804.10959/) (2018)
- [Joint CTC-Attention based End-to-End Speech Recognition using Multi-task Learning](https://mlsanity.com/paper/1609.06773/) (2016)
- [End-to-End Attention-based Large Vocabulary Speech Recognition](https://mlsanity.com/paper/1508.04395/) (2015)
- [Self-Training for End-to-End Speech Recognition](https://mlsanity.com/paper/1909.09116/) (2019)
- [Iterative Pseudo-Labeling for Speech Recognition](https://mlsanity.com/paper/2005.09267/) (2020)
- [Multilingual and code-switching ASR challenges for low resource Indian languages](https://mlsanity.com/paper/2104.00235/) (2021)
- [On the End-to-End Solution to Mandarin-English Code-switching Speech Recognition](https://mlsanity.com/paper/1811.00241/) (2018)
- [Reducing Language confusion for Code-switching Speech Recognition with Token-level Language Diarization](https://mlsanity.com/paper/2210.14567/) (2022)
- [Multi-Encoder-Decoder Transformer for Code-Switching Speech Recognition](https://mlsanity.com/paper/2006.10414/) (2020)
- [Transformer-Transducers for Code-Switched Speech Recognition](https://mlsanity.com/paper/2011.15023/) (2020)
- [Pseudo-Labeling for Massively Multilingual Speech Recognition](https://mlsanity.com/paper/2111.00161/) (2021)
- [Language-specific Characteristic Assistance for Code-switching Speech Recognition](https://mlsanity.com/paper/2206.14580/) (2022)
- [Semi-Supervised Learning with Data Augmentation for End-to-End ASR](https://mlsanity.com/paper/2007.13876/) (2020)
- [Towards End-to-End Code-Switching Speech Recognition](https://mlsanity.com/paper/1810.13091/) (2018)
- [Towards Zero-Shot Code-Switched Speech Recognition](https://mlsanity.com/paper/2211.01458/) (2022)
- [Aligning Speech to Languages to Enhance Code-switching Speech Recognition](https://mlsanity.com/paper/2403.05887/) (2024)
- [Continuous Pseudo-Labeling from the Start](https://mlsanity.com/paper/2210.08711/) (2022)
- [Mandarin-English Code-switching Speech Recognition with Self-supervised Speech Representation Models](https://mlsanity.com/paper/2110.03504/) (2021)
- [A Language Agnostic Multilingual Streaming On-Device ASR System](https://mlsanity.com/paper/2208.13916/) (2022)
- [Continuous Soft Pseudo-Labeling in ASR](https://mlsanity.com/paper/2211.06007/) (2022)

## Abstract (arXiv)

We determine exactly what a kurtosis bound buys for one-sided tail control. For the class C(kappa) of real random variables with mean 0, variance 1, and fourth moment at most kappa, the skewness left free, we compute the worst-case tail probability V1(t,kappa)=supX∈C(kappa)mathbbP(X≥ t) for every threshold t>0 and every kappa≥ 1. The answer is a four-regime map: a Cantelli tongue b(kappa)≤ t≤ c(kappa) on which the two-moment bound 1/(1+t2) remains tight and the kurtosis constraint is worthless; a tail regime t≥ c(kappa) with the closed form V1=(kappa-1)/((t2-1)2+kappa-1); a plateau regime, present only for kappa≤ 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(kappa) 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, Zhang, and Zhang (2010) at t=0.