arXiv subjects

Yi-Kun Zhao

Publications and source records attributed to Yi-Kun Zhao.

3 recordsLinked to original sources

arXiv2026

Central limit theorem for high temperature spin models via martingale embedding

We use martingale embeddings to prove a central limit theorem (CLT) for one-dimensional projections of high-dimensional random vectors in $\{-1,1\}^n$ satisfying a Poincaré inequality. We obtain a non-asymptotic error bound involving two-point and three-point functions for the CLT in 2-Wasserstein distance. We present three illustrative applications: Ising model with finite-range interactions, ferromagnetic Ising model under the Dobrushin condition, and the Sherrington-Kirkpatrick spin glass model at sufficiently high temperature. In all the examples, we allow heterogeneous external fields.

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↗

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↗