OP-00757 Solution claimed

A uniform Gaussian approximation bound for Jack measures

For an integer $n \ge 3$ and $\alpha \ge 1$, let $\lambda \vdash n$ be drawn from the Jack measure $$\mathbb P_\alpha(\lambda) = \frac{\alpha^n n!}{\prod_{u \in \lambda} \bigl(\alpha a(u) + \ell(u) + 1\bigr)\bigl(\alpha a(u) + \ell(u) + \alpha\bigr)},$$ where $a(u)$ and $\ell(u)$ count the boxes to the right of and below the box $u$ in the Young diagram of $\lambda$; for $\alpha = 1$ this is the Plancherel measure. Write $\lambda_i$ and $\lambda_i'$ for the row and column lengths of $\lambda$, and let $$W_{n,\alpha} = \frac{\sum_i \Bigl[\alpha \binom{\lambda_i}{2} - \binom{\lambda_i'}{2}\Bigr]}{\sqrt{\alpha \binom{n}{2}}},$$ which for $\alpha = 1$ is the normalized character ratio of a transposition. Its third moment is $(\alpha - 1)/\sqrt{\alpha \binom{n}{2}}$, so $\sqrt\alpha/n \to 0$ is necessary for asymptotic normality (Fulman). Fulman and Goldstein proved a bound of order $\max\{n^{-1/2}, \sqrt\alpha/n\}$ with an absolute constant for the $L^1$ (Wasserstein) distance between the law of $W_{n,\alpha}$ and the standard normal law. In the Kolmogorov distance, Chen, Raič and Thành proved $$\sup_{x \in \mathbb R} \bigl| \mathbb P_\alpha(W_{n,\alpha} \le x) - \Phi(x) \bigr| \le 8.2 \max\Bigl\{ \frac{1}{\sqrt n}, \frac{\sqrt\alpha \log n}{n} \Bigr\}$$ for all $n \ge 3$ and $\alpha \ge 1$, where $\Phi$ is the standard normal distribution function.

Problem. Is there a universal constant $C < \infty$, independent of both $n$ and $\alpha$, such that $$\sup_{x \in \mathbb R} \bigl| \mathbb P_\alpha(W_{n,\alpha} \le x) - \Phi(x) \bigr| \le C \max\Bigl\{ \frac{1}{\sqrt n}, \frac{\sqrt\alpha}{n} \Bigr\}$$ for all integers $n \ge 3$ and all $\alpha \ge 1$? A constant depending on a fixed $\alpha$ does not suffice.

Source: Fulman (2004), §1, the conjecture after his main theorem, for α ≥ 1. Stated for n ≥ 3 by AIM — Open Applied Problems, after Chen, Raič and Thành (2021).

Imported from github.com/MColbrook/AIM.

Classification

Primary: PR — Probability
Secondary: CO
Keywords: AIM Open Applied Problems, Berry–Esseen bounds, Jack measure, Stein's method

References

@article{fulman2004,
  author        = {Fulman, Jason},
  title         = {Stein's method, {J}ack measure, and the {M}etropolis algorithm},
  journal       = {J. Combin. Theory Ser. A},
  volume        = {108},
  number        = {2},
  year          = {2004},
  pages         = {275--296},
  doi           = {10.1016/j.jcta.2004.07.003},
  eprint        = {math/0311290},
  archivePrefix = {arXiv},
}
@article{chen2021,
  author        = {Chen, Louis H. Y. and Rai{\v{c}}, Martin and Th{\`a}nh, L{\^e} V{\u{a}}n},
  title         = {On the error bound in the normal approximation for {J}ack measures},
  journal       = {Bernoulli},
  volume        = {27},
  number        = {1},
  year          = {2021},
  pages         = {442--468},
  doi           = {10.3150/20-BEJ1245},
  eprint        = {1902.03476},
  archivePrefix = {arXiv},
}
@article{fulman2011,
  author        = {Fulman, Jason and Goldstein, Larry},
  title         = {Zero biasing and {J}ack measures},
  journal       = {Combin. Probab. Comput.},
  volume        = {20},
  number        = {5},
  year          = {2011},
  pages         = {753--762},
  doi           = {10.1017/S0963548311000241},
  eprint        = {1010.4759},
  archivePrefix = {arXiv},
}
@misc{aim-jack-normal,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A uniform Gaussian approximation bound for Jack measures}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/636-jack-normal.md},
  note         = {Entry 636 at commit a153f46, 2026-10-13},
}

Resolution claims

A uniform Kolmogorov bound for Jack measures v1 GitHub ↗

Kaiwen Jin

For the normalized transposition statistic under Jack measure on partitions of $n$, we prove a Kolmogorov bound $10\max\{n^{-1/2},\sqrt\alpha/n\}$ for every $n\ge3$ and $\alpha\ge1$. This removes the logarithmic loss in the known bound, including the transition $\alpha\asymp n$, and proves the uniform estimate conjectured by Fulman. The proof retains cancellation in the conditional third moment of a Jack growth exchangeable pair. …

Full resolution · not verified · fidelity not established · posted on GitHub 13 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, paper/jack_normal.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00757, "A uniform Gaussian approximation bound for Jack measures", clopen.solutions.