Stein's Method, Self-normalized Limit Theory and Applications · 2010-prob-184

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. No citable proof or counterexample was found for the stated uniform, distribution-free remainder bound. The available literature reaches the same leading term and the optimal moderate-deviation range, but not the asserted quantitative error.

Qi-Man Shao, “Stein's Method, Self-normalized Limit Theory and Applications,” Proceedings of the International Congress of Mathematicians 2010, Vol. IV, 2010, pp. 2325–2358 (direct proceedings PDF), states the displayed assertion as Conjecture 1. It is modeled on the nonmaximal rate established by Bing-Yi Jing, Qi-Man Shao, and Qiying Wang, “Self-normalized Cramér-type large deviations for independent random variables,” The Annals of Probability 31 (2003), 2167–2215, doi:10.1214/aop/1068646382. The latter paper proves the analogous explicit relative-error bound for $\mathbb P(S_n\geq xV_n)$, with leading constant $1$, but it does not address $\max_{k\leq n}S_k$.

Zhishui Hu, Qi-Man Shao, and Qiying Wang, “Cramér Type Moderate deviations for the Maximum of Self-normalized Sums,” Electronic Journal of Probability 14 (2009), 1181–1197, doi:10.1214/EJP.v14-663, prove

uniformly for $0\leq x=o(n^{1/6})$, under the stronger assumption $\mathbb E X^4<\infty$. This establishes the leading term only and neither gives the conjectured rate nor works under the full $(2+r)$-moment hypothesis.

Weidong Liu, Qi-Man Shao, and Qiying Wang, “Self-normalized Cramér type moderate deviations for the maximum of sums,” Bernoulli 19 (2013), 1006–1027, doi:10.3150/12-BEJ415, arXiv:1307.6044, provide the strongest direct partial result. Their Theorem 1 proves convergence to $2$ uniformly for $0\leq x=o(n^{r/(4+2r)})$ under finite $(2+r)$-moments (and even treats independent, non-identically distributed variables under uniform moment bounds). Their more general theorem and Proposition 1 give other explicit errors, but not the error in the present problem and not uniformly through its stated endpoint. Most decisively, Remark 1 explicitly repeats the requested quantitative estimate for $r=1$ as a conjecture. Its factor $1+x^3$ is equivalent up to universal constants to $(1+x)^3$, so even this special case of the present assertion was not resolved there.

Thus the established progress is the sharp $2+o(1)$ moderate-deviation theorem under the required moment assumptions. What remains open is the absolute-constant bound of order

uniformly over the entire stated range, already including the explicitly re-proposed case $r=1$. Searches of later papers citing the 2013 result, including work on self-normalized processes, dependent sequences, and martingales, found no theorem closing this maximal-sum rate gap.