arXiv subjects

Song-Hao Liu

Publications and source records attributed to Song-Hao Liu.

At least 19 recordsLinked to original sources

arXiv2026

Sharp moderate deviations and uniform Berry--Esseen bounds for Hotelling's $T^2$-statistic via Stein's method

Let $\mathbf X_1,\ldots,\mathbf X_n$ be independent random vectors in $\mathbb R^d$ with common mean $\boldsymbolμ_n$, where $d\ge2$ is a fixed integer. Let $\bar{\mathbf X}_n=n^{-1}\sum_{i=1}^n\mathbf X_i$ denote the sample mean and let \[ \widehatΣ_n=\frac1n\sum_{i=1}^n (\mathbf X_i-\bar{\mathbf X}_n) (\mathbf X_i-\bar{\mathbf X}_n)^\top \] denote the sample covariance matrix. Assume that the average covariance matrix is positive definite and that $\widehatΣ_n$ is nonsingular almost surely. For testing $H_0:\boldsymbolμ_n=\boldsymbolμ_0$, the Hotelling statistic is \[ T^2_{H,n} =n(\bar{\mathbf X}_n-\boldsymbolμ_0)^\top \widehatΣ_n^{-1} (\bar{\mathbf X}_n-\boldsymbolμ_0). \] Under $H_0$, we establish finite-sample Cramér-type moderate deviation results and uniform Berry--Esseen bounds under finite $p$th moments for $2<p\le3$. The moderate deviation error order is optimal over independent triangular arrays, as shown by explicit constructions. For identically distributed observations with a fixed distribution, we obtain \[ \sup_{0\le x\le b_n} \left| \frac{\mathbb P(T^2_{H,n}\ge x^2)} {\mathbb P(χ_d^2\ge x^2)}-1 \right| = O\!\left((1+b_n)^p n^{-(p-2)/2}\right) \] for every nonnegative sequence $b_n=o(n^{(p-2)/(2p)})$. The uniform Berry--Esseen bound is $O(n^{-(p-2)/2})$. The proof uses a radial Stein identity and a recursive error estimate. We apply the results to chi-squared calibration in simultaneous testing.

math.ST↗

arXiv2026

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

For a rowwise independent triangular array $\{X_{n,i}:1\le i\le n\}$, put $S_n=\sum_{i=1}^nX_{n,i}$ and $V_n^2=\sum_{i=1}^nX_{n,i}^2$. We prove that $S_n/V_n\xrightarrow{\mathrm d}{\mathrm N}(0,1)$ implies $\max_{i\le n}|X_{n,i}|/V_n\xrightarrow{\mathbb P}0$. The same assumption gives $V_n^2/b_n^2\xrightarrow{\mathbb P}1$ for any choice of medians $b_n^2$ of $V_n^2$. Combined with the results of Shao (2018), this yields necessary and sufficient conditions for the self-normalized central limit theorem and resolves his conjecture for independent sequences. The proof identifies the joint limits of the sum and its sum of squares, then uses the standard normal ratio to remove every non-Gaussian component.

math.PR↗

arXiv2026

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

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 $n^{-1}$. At the fixed critical parameters $(α_c,h_c),$ the centered fluctuations satisfy $n^{2/3}(Q-\mathbb{E} Q)\Rightarrow \mathcal{N}(0,σ_0^2)$, with explicit variance $σ_0^2>0$. This disproves the quartic fluctuation conjecture of Bianchi, Collet and Magnanini (2024). Gaussian limits also hold $(α_c+ λ/n,h_c + γ/n)$, except when the coefficients satisfy $-\frac{11}{16}+\frac{2}{9}(γ+\frac49λ)=0$. Under this linearity condition, the limiting laws of $\sqrt n(Q-2/3)$ have densities proportional to $\exp\{-81z^4/64+9θz^2/8\}$, where $θ=-57/16+8λ/27$. We obtain a sharp $n^{-1/3}$ rate in $1$-Wasserstein distance for the Gaussian limits and an $O(n^{-1/2})$ bound for the quartic limits. The proofs combine Stein's method for exchangeable pairs with conditional row estimates and a local Dobrushin contraction.

math.PR↗

arXiv2026

Three quantitative central limit theorems for the unbounded excursion component of a Gaussian field

For a class of smooth stationary Gaussian fields, we study the fluctuations of the volume, surface area and Euler characteristic of the unbounded excursion component in growing boxes $Λ_n=[-n,n]^d$. We establish nondegenerate central limit theorems for all three functionals throughout the supercritical regime in every dimension $d\geq 2$. We further quantify these central limit theorems by establishing Berry--Esseen bounds of order $O\bigl(|Λ_n|^{-(1-1/s)}\bigr)$ in Kolmogorov distance, where $s\in(1,2]$ depend on the regularity and decay assumptions. Especially for the Bargmann--Fock field, we obtain a Berry--Esseen bound of $O\bigl(|Λ_n|^{-1/2}\bigr)$ for all three functionals. A matching lower bound shows that this rate is sharp for the Euler characteristic. For the volume, we improve the previously known rate of $O\bigl(|Λ_n|^{-1/2}(\log|Λ_n|)^{2d}\bigr)$. We mainly use Stein's method and moment bounds for differences arising from white-noise resampling to finish our proof.

math.PR↗

arXiv2026

Refined Cramér-type moderate deviations by Stein's method under dependence

In this paper, we establish refined Cramér-type moderate deviations for dependent random variables at any fixed order $m\geq3$. Combining higher-order Stein identities with exponential tilting, we obtain cumulant-corrected tail approximations with explicit error bounds for boundedly supported kernels. Our main application gives a fourth-order expansion under bounded local dependence, reaching an $O(n^{3/10})$ deviation range under standard normalization with fixed dependence parameters. It improves on previous work in both deviation range and applicability, and also covers bounded $m$-dependent sequences without an independent-base representation. Further applications to the generalized combinatorial central limit theorem and the Curie-Weiss model also extend some earlier results under permutation and mean-field dependence.

math.PR↗

arXiv2026

Fluctuations of subgraph counts in exponential random graphs

In this paper, we study fluctuations of fixed subgraph counts in dense exponential random graphs on $n$ vertices. We compare each normalized subgraph count $V$ with the normalized edge count $W$: their difference $V-W$ is $O(n^{-1/2})$ in $L^2$, whereas $\mathbb{E}[(V-W)f(W)]=O(n^{-1})$ for every $1$-Lipschitz function $f$. We also obtain $O(n^{-1})$ Kolmogorov and Wasserstein bounds, optimal for the edge count, with relative variance error of the same order. These estimates hold uniformly on compact subsets of the ferromagnetic subcritical and absolute-coefficient Dobrushin regions, and under small fixed signed perturbations of the former. Using this uniform control, we derive a local Gaussian likelihood expansion with rank-one leading Fisher information and two-sided moderate deviations with an explicit cubic correction. The proofs combine two applications of the normal Stein equation, covariance estimates from Glauber dynamics and exponential tilting.

math.PR↗

arXiv2026

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

We establish Berry--Esseen bounds and 1-Wassershtein bounds for the total spin of strictly subcritical zero-field Ising models with summable interactions. Let $J:\mathbb Z^d\to[0,\infty)$ be symmetric, satisfy $J(0)=0$, and $\|J\|_1:=\sum_{x\in\mathbb Z^d}J(x)<\infty$. For a nonempty finite set $Λ\subset\mathbb Z^d$, consider zero-field Gibbs measure with free boundary conditions, in which a spin configuration $σ\in\{-1,+1\}^Λ$ has probability proportional to \[ \exp\!\left\{ β\sum_{x,y\inΛ} J(x-y)σ_xσ_y \right\}. \] Let $S_Λ:=\sum_{x\inΛ}σ_x$ be the total spin and $β_{\mathrm c}$ denote the critical inverse temperature. For every fixed $0\leqβ<β_{\mathrm c}$, we prove that \[ d_{\mathrm K}\!\left( \frac{S_Λ}{\sqrt{\operatorname{Var}(S_Λ)}},G \right) \leq C_{β,J}|Λ|^{-1/2}, \] where $d_{\mathrm K}$ is the Kolmogorov distance, $G\sim \mathcal{N}(0,1)$, and $C_{β,J}$ depends only on $β$ and $J$. For the fully periodized model on $(\mathbb Z/L\mathbb Z)^d$, the corresponding rate is $L^{-d/2}$, uniformly in $L$. For the classical nearest-neighbor model, the Kolmogorov bound also holds for $-β_{\mathrm c}<β<β_{\mathrm c}$, covering both ferromagnetic and antiferromagnetic interactions. We also obtain $1$-Wasserstein bounds with the same rates in the respective subcritical regimes. Both the Kolmogorov and $1$-Wasserstein bounds hold with the same rate under the corresponding infinite-volume free-boundary Gibbs measures.

math.PR↗

arXiv2026

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

Let $X_1,X_2,\ldots$ 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-normalized maximum of partial sums under a finite $p$-th absolute moment, $2<p\le3$. In particular, we show that $\mathbb{P}(M_n\ge xV_n)=2(1-Φ(x))\{1+O(1)β_{n,p}(1+x)^p\}$ for $0\le x\leβ_{n,p}^{-1/p}$, where $M_n=\max_{1\le k\le n}\sum_{j=1}^kX_j$, $V_n=(\sum_{j=1}^nX_j^2)^{1/2}$, $β_{n,p}=\mathbb{E}|X_1|^p/(n^{(p-2)/2}σ^p)$, and $Φ$ is the standard normal distribution function. This proves a conjecture of Shao2010. More precisely, we bound the relative error by the same truncated-moment quantity as for self-normalized sums. The proof compares the maximum with the absolute sum while preserving their common denominator, through a reflected-walk representation and Fourier estimates with distribution-adapted smoothing.

math.PR↗

arXiv2026

Mixing Time of Conditional Two Star Exponential Random Graphs

Two star exponential random graph models (ERGMs) are an interesting special case of both general ERGMs and mean-field Ising models. In this paper, we study two star ERGMs conditioning on the edge density $p$. We begin with an analytic characterization of the replica symmetric region, where the conditional model is close in cut distance to the Erdős--Rényi random graph $G(n,p)$. We prove that this region is twice as large as the corresponding region for the unconditional model. To study refined properties of the conditional model, we then analyze the global Kawasaki algorithm for sampling from it. Within the replica symmetric region, and under the additional condition that $4βp(1-p)<0.5$ when $|p-1/2|\lesssim 0.4632$, where $β$ is the model parameter, we prove metastable fast mixing of the Kawasaki algorithm. As corollaries, we obtain a weak Poincaré inequality and use it to deduce concentration inequalities of optimal order for conditional subgraph counts. The additional condition $4βp(1-p)<1/2$ if $|p-1/2|\lesssim 0.4632$ comes from our proof technique of contractive coupling. This bottleneck did not appear in previous studies applying the contractive coupling technique to unconditional ERGMs.

math.PR↗

arXiv2026

Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials

We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials $P_n$, where $n$ denotes the degree and is assumed to be sufficiently large. For each fixed $B$ above an absolute threshold, let $I_n=[-\sqrt n+B\sqrt{\log n}, \sqrt n-B\sqrt{\log n}]$. Uniformly over deterministic compact intervals $I\subseteq I_n$ whose length $\ell$ is sufficiently large and depends on $n$, the distribution of the standardized number of real roots in $I$ has Kolmogorov distance at most $C\log\ell/\sqrt\ell$ from the standard Gaussian distribution. Consequently, every such interval sequence with $\ell\to\infty$ satisfies a central limit theorem. In particular, taking $I=I_n$ gives the bound $C\log n/n^{1/4}$. The same bound holds for the distribution of the standardized number of real roots on $\mathbb R$. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.

math.PR↗

arXiv2026

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

Let $A=(A_n)_{n\ge2}$ be a triangular array of random matrices, where $A_n=(a_{ij})_{1\le i,j\le n}$ is an $n\times n$ random matrix with independent real entries satisfying $\mathbb E a_{ij}=0$ and $\mathbb Ea_{ij}^2=1$, and put $\mathcal L_n=\log|\det A_n|$ and \[ W_n^{\mathrm d}(A_n):=\frac{\mathcal L_n - \frac12\log(n-1)!}{\sqrt{\frac12\log n}},\quad W_n^{\mathrm e}(A_n):= \frac{\mathcal L_n-\mathbb E \mathcal L_n}{\sqrt{\frac12\log n}}. \] We prove that $W_n^{\mathrm d}(A_n) \Rightarrow \mathcal N(0,1)$, whenever the family $\left\{\frac{|a_{ij}|^{4}}{\sqrt{\log(e+|a_{ij}|)}} \right\}_{n\geq 2;1\leq i,j\leq n}$ is uniformly integrable. If, in addition, the entries have uniformly bounded densities, then $W_n^{\mathrm e}(A_n) \Rightarrow \mathcal N(0,1)$ whenever the family $\left\{\frac{|a_{ij}|^{4}}{\log(e+|a_{ij}|)}\right\}_{n\geq 2;1\leq i,j\leq n} $ is uniformly integrable. These two conditions are optimal at the level of universal moment assumptions. We further establish the corresponding Berry--Esseen bounds, and show that for $0<δ\le\tfrac12$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1/2-δ}}<\infty$, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm d}(A_n),\mathcal N(0,1))\le C(\log n)^{-δ}. \end{align*} For $0<γ\le1$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1-γ}}<\infty$ and the entries have uniformly bounded densities, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm e}(A_n),\mathcal N(0,1))\le C(\log n)^{-γ}. \end{align*} When $δ= 1/2$ and $γ= 1$, the bounds $(\log n)^{-1/2}$ and $(\log n)^{-1}$ are optimal, respectively. Our results improve the earlier Central Limit Theorem by \cite{BaoPanZhou2015} and the Berry--Esseen bound by \cite{NguyenVu2014}.

math.PR↗

arXiv2026

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

We establish free-energy fluctuation limits for the Ising Sherrington--Kirkpatrick model in the nonzero parts of its critical window. For fixed $b\ne0$ and $β_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}G+\sqrt{\frac32}\,b_+TW_1, \] where $G$ is standard Gaussian, $TW_1$ has the real Tracy--Widom law, and $G$ is independent of $TW_1$, and $\mathrm{FE}$ denotes the limit of spherical Sherrington--Kirkpatrick free-energy. Additionally, we show that in a moderately supercritical regime \[ \frac{2}{N^{1/3}(β_N-1)}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}TW_1. \] We also show that the above results remains valid for independent, not necessarily identically distributed, disorder matrices whose first three moments match the Gaussian law and whose fourth moments satisfy an averaged bound.

math.PR↗

arXiv2025

Conditional central limit theorems for exponential random graphs

In this paper, we study the Exponential Random Graph Models (ERGMs) conditioning on the number of edges. In subcritical region of model parameters, we prove a conditional Central Limit Theorem (CLT) with explicit mean and variance for the number of two stars. This generalizes the corresponding result in the literature for the Erdős--Rényi random graph. To prove our main result, we develop a new conditional CLT via exchangeable pairs based on the ideas of Dey and Terlov. Our key technical contributions in the application to ERGMs include establishing a linearity condition for an exchangeable pair involving two star counts, a local CLT for edge counts, as well as new higher-order concentration inequalities. Our approach also works for general subgraph counts, and we give a conjectured form of their conditional CLT.

math.PR↗

arXiv2025

Normal approximation for exponential random graphs

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 remained an open problem. In this paper, we establish the CLT. As a result of our proof, we also derive a convergence rate for the CLT, an explicit formula for the asymptotic variance, and the CLT for general subgraph counts. To establish our main result, we develop Stein's method for the normal approximation of general functionals of nonlinear exponential families of random variables, which is of independent interest. In addition to ERGMs, our general theorem can also be applied to other models. A key ingredient needed in our proof for the ERGM is a higher-order concentration inequality, which was known in a subset of the subcritical region called Dobrushin's uniqueness region. We use Stein's method to partially generalize such inequalities to the subcritical region.

math.PR↗

arXiv2024

Symmetric KL-divergence by Stein's Method

In this paper, we consider the symmetric KL-divergence between the sum of independent variables and a Gaussian distribution, and obtain a convergence rates of order $O\left( \frac{\ln n}{\sqrt{n}}\right)$. The proof is based on Stein's method. The convergence rate of order $O\left( \frac{1}{\sqrt{n}}\right)$ and $O\left( \frac{1}{n}\right) $ are also obtained under higher moment condition.

math.PR↗

arXiv2023

High-dimensional Central Limit Theorems by Stein's Method in the Degenerate Case

In the literature of high-dimensional central limit theorems, there is a gap between results for general limiting correlation matrix $Σ$ and the strongly non-degenerate case. For the general case where $Σ$ may be degenerate, under certain light-tail conditions, when approximating a normalized sum of $n$ independent random vectors by the Gaussian distribution $N(0,Σ)$ in multivariate Kolmogorov distance, the best-known error rate has been $O(n^{-1/4})$, subject to logarithmic factors of the dimension. For the strongly non-degenerate case, that is, when the minimum eigenvalue of $Σ$ is bounded away from 0, the error rate can be improved to $O(n^{-1/2})$ up to a $\log n$ factor. In this paper, we show that the $O(n^{-1/2})$ rate up to a $\log n$ factor can still be achieved in the degenerate case, provided that the minimum eigenvalue of the limiting correlation matrix of any three components is bounded away from 0. We prove our main results using Stein's method in conjunction with previously unexplored inequalities for the integral of the first three derivatives of the standard Gaussian density over convex polytopes. These inequalities were previously known only for hyperrectangles. Our proof demonstrates the connection between the three-components condition and the third moment Berry--Esseen bound.

math.PR↗

arXiv2022

Edgeworth Expansion by Stein's Method

Edgeworth expansion provides higher-order corrections to the normal approximation for a probability distribution. The classical proof of Edgeworth expansion is via characteristic functions. As a powerful method for distributional approximations, Stein's method has also been used to prove Edgeworth expansion results. However, these results assume that either the test function is smooth (which excludes indicator functions of the half line) or that the random variables are continuous (which excludes random variables having only a continuous component). Thus, how to recover the classical Edgeworth expansion result using Stein's method has remained an open problem. In this paper, we develop Stein's method for two-term Edgeworth expansions in a general case. Our approach involves repeated use of Stein equations, Stein identities via Stein kernels, and a replacement argument.

math.PR↗

arXiv2021

Cramér-type Moderate deviations under local dependence

We establish Cramér-type moderate deviation theorems for sums of locally dependent random variables and combinatorial central limit theorems. Under some mild exponential moment conditions, optimal error bounds and convergence ranges are obtained. Our main results are more general or shaper than the existing results in the literature. The main results follows from a more general Cramér-type moderate deviation theorem for dependent random variables without any boundedness assumptions, which is of independent interest. The proofs couple Stein's method with a recursive argument.

math.PR↗