OP-00707 Solution claimed

Small-ball ratios for general symmetric product priors

Fix $1 \le p < \infty$ and positive sequences $\alpha = (\alpha_k)$ and $\gamma = (\gamma_k)$, and let $$X = \Bigl\{x \in \mathbb{R}^{\mathbb{N}} : \|x\|_X = \Bigl(\sum_{k=1}^\infty |x_k/\alpha_k|^p\Bigr)^{1/p} < \infty\Bigr\}.$$ Let $\rho$ be a continuous, even probability density on $\mathbb{R}$, strictly decreasing on $[0, \infty)$ (so $\rho > 0$ everywhere), and fix $m \in X$. For independent random variables $Z_k$ with density $\rho$, let $\mu$ be the law of $(m_k + \gamma_k Z_k)_{k \ge 1}$, and assume $\mu(X) = 1$. For $h \in X$ put $$Q(h) = \sum_{k=1}^\infty \log \frac{\rho(0)}{\rho((h_k - m_k)/\gamma_k)} \in [0, \infty],$$ and let $B_r(h) = \{x \in X : \|x - h\|_X < r\}$. Ayanbayev, Klebanov, Lie and Sullivan proved that the upper limit of $\mu(B_r(h))/\mu(B_r(m))$ as $r \downarrow 0$ is at most $e^{-Q(h)}$ for every $h \in X$, where $e^{-\infty} = 0$, and that the ratio converges to $e^{-Q(h)}$ when, in addition, $\rho \in C^2$ with $\rho'' \in L^1$, $Q(h) < \infty$ and $\sum_k ((h_k - m_k)/\gamma_k)^2 < \infty$.

Problem. Is it true that $$\lim_{r \downarrow 0} \frac{\mu(B_r(h))}{\mu(B_r(m))} = e^{-Q(h)} \qquad \text{for every } h \in X,$$ where $e^{-\infty} = 0$? No differentiability, log-concavity or finite Fisher information assumption is made on $\rho$.

Source: Ayanbayev, Klebanov, Lie and Sullivan (2022), Conjecture 4.12, under their Assumption 4.1 (A1)–(A3).

Imported from github.com/MColbrook/AIM.

Classification

Primary: PR — Probability
Secondary: ST
Keywords: AIM Open Applied Problems, Onsager–Machlup functionals, product measures, small-ball probabilities

References

@article{ayanbayev2022ii,
  author        = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, T. J.},
  title         = {{$\Gamma$}-convergence of {O}nsager--{M}achlup functionals: {II}. {I}nfinite product measures on {B}anach spaces},
  journal       = {Inverse Problems},
  volume        = {38},
  number        = {2},
  year          = {2022},
  pages         = {025006},
  doi           = {10.1088/1361-6420/ac3f82},
  eprint        = {2108.04598},
  archivePrefix = {arXiv},
}
@misc{aim-product-prior-small-ball-ratios,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Small-ball ratios for general symmetric product priors}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/660-product-prior-small-ball-ratios.md},
  note         = {Entry 660 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

A quartic product-prior counterexample to the general small-ball ratio conjecture v1 GitHub ↗

Siavash Sadeghi

We give a proposed counterexample to the unresolved general statement in AIM Problem 506, equivalently Conjecture 4.12 of Ayanbayev, Klebanov, Lie and Sullivan. On $\ell^2$ we construct a countable product of scaled copies of the density $\rho(z)\propto e^{-z^4}$ and an explicit center $h$ with formal energy $Q(h)=1$, but \[ _{r0}\frac{(B_r(h))}{(B_r(0))}=0 e^{-1}. \] The density is positive, real analytic and log-concave, …

Counterexample · not verified · fidelity not established · posted on GitHub 2 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, siavash-sadeghi-506/aim506_counterexample.pdf.

Citing this problem

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

OP-00707, "Small-ball ratios for general symmetric product priors", clopen.solutions.