Qi-Man Shao

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math.PR2026

Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays

Xiang Li, Song-Hao Liu, Qi-Man Shao +2

For a rowwise independent triangular array ${X_{n, i} : 1 \leq i \leq n}$ , put $S_{n} = \sum_{i = 1}^{n} X_{n, i}$ and $V_{n}^{2} = \sum_{i = 1}^{n} X_{n, i}^{2}$ . We prove that $S_n/V_n\xrightarrow{\mathrm d}…

math.PR2026

Critical fluctuations in the edge--triangle exponential random graph models

Song-Hao Liu, Qi-Man Shao, Hao Shi +1

We prove quantitative limit theorems for the edge density $Q$ in the dense edge--triangle exponential random graph model at criticality and under parameter perturbations of order $…

math.PR2026

Normal approximation for total magnetization in subcritical long-range Ising models

Song-Hao Liu, Qi-Man Shao, Yang Xie +1

We establish Berry--Esseen bounds and 1-Wassershtein bounds for the total spin of strictly subcritical zero-field Ising models with summable interactions. Let $J:\mathbb Z^d\to[0,\…

math.PR2026

Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums

Song-Hao Liu, Qi-Man Shao, Hao Shi +1

Let $X_{1}, X_{2}, \dots$ be i.i.d. random variables with zero mean and variance $σ^{2} \in (0, \infty)$ . In this paper, we establish a sharp Cramér-type moderate deviation for the self-norm…

math.PR2026

The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants

Song-Hao Liu, Qi-Man Shao, Jing-Yu Xu

Let $A = (A_{n})_{n \geq 2}$ be a triangular array of random matrices, where $A_{n} = (a_{ij})_{1 \leq i, j \leq n}$ is an $n \times n$ random matrix with independent real entries satisfying $\math…

math.PR2026

Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model

Yu Cheng, Song-Hao Liu, Qi-Man Shao +1

We establish free-energy fluctuation limits for the Ising Sherrington--Kirkpatrick model in the nonzero parts of its critical window. For fixed $b \neq = 0$ and $β_N=1+bN^{-1/3}\sqrt{\l…

math.PR2026

Cramér-type moderate deviation for double index permutation statistics

Songhao Liu, Qiman Shao, Jingyu Xu

We establish a Cramér-type moderate deviation theorem for double-index permutation statistics (DIPS). To the best of our knowledge, previous results only provided Berry-Esseen type…

math.PR2024

Normal approximation for exponential random graphs

Xiao Fang, Song-Hao Liu, Qi-Man Shao +1

The question of whether the central limit theorem (CLT) holds for the total number of edges in exponential random graph models (ERGMs) in the subcritical region of parameters has r…