arXiv subjects

Zhonggen Su

Publications and source records attributed to Zhonggen Su.

24 records · Page 2Linked to original sources

arXiv2023

Rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales

We give some rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales with differences having finite variances. For the Kolmogorov distances, we present some exact Berry-Esseen bounds for martingales, which generalizes some Berry-Esseen bounds due to Bolthausen. For the Wasserstein distance, with Stein's method and Lindeberg's telescoping sum argument, the rates of convergence in martingale central limit theorems recover the classical rates for sums of i.i.d.\ random variables, and therefore they are believed to be optimal.

math.PR↗

arXiv2023

Tail Bounds on the Spectral Norm of Sub-Exponential Random Matrices

Let $X$ be an $n\times n$ symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of $X$ with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of $X$ with sub-Exponential entries. Our method relies upon a crucial ingredient of a novel chaining argument that essentially involves both the particular structure of the sets used for the chaining and the distribution of coordinates of a point on the unit sphere.

math.PR↗

arXiv2019

A Donsker-type Theorem for Log-likelihood Processes

Let $(Ω, \mathcal{F}, (\mathcal{F})_{t\ge 0}, P)$ be a complete stochastic basis, $X$ a semimartingale with predictable compensator $(B, C, ν)$. Consider a family of probability measures $\mathbf{P}=( {P}^{n, ψ}, ψ\in Ψ, n\ge 1)$, where $Ψ$ is an index set, $ {P}^{n, ψ}\stackrel {loc} \ll{P}$, and denote the likelihood ratio process by $Z_t^{n, ψ} =\frac{dP^{n, ψ}|_{\mathcal{F}_t}}{d P|_{\mathcal{F}_t}}$. Under some regularity conditions in terms of logarithm entropy and Hellinger processes, we prove that $\log Z_t^{n}$ converges weakly to a Gaussian process in $\ell^\infty(Ψ)$ as $n\rightarrow\infty$ for each fixed $t>0$.

math.PR↗

arXiv2017

On Bernstein Type Inequalities for Stochastic Integrals of Multivariate Point Processes

We consider the stochastic integrals of multivariate point processes and study their concentration phenomena. In particular, we obtain a Bernstein type of concentration inequality through Doléans-Dade exponential formula and a uniform exponential inequality using a generic chaining argument. As applications, we obtain a upper bound for a sequence of discrete time martingales indexed by a class of functionals, and so derive the rate of convergence for nonparametric maximum likelihood estimators, which is an improvement of earlier work of van de Geer.

math.PR↗

arXiv2014

Joint CLT for several random sesquilinear forms with applications to large-dimensional spiked population models

In this paper, we derive a joint central limit theorem for random vector whose components are function of random sesquilinear forms. This result is a natural extension of the existing central limit theory on random quadratic forms. We also provide applications in random matrix theory related to large-dimensional spiked population models. For the first application, we find the joint distribution of grouped extreme sample eigenvalues correspond to the spikes. And for the second application, under the assumption that the population covariance matrix is diagonal with $k$ (fixed) simple spikes, we derive the asymptotic joint distribution of the extreme sample eigenvalue and its corresponding sample eigenvector projection.

math.PR↗

arXiv2011

Local Semicircle law and Gaussian fluctuation for Hermite $β$ ensemble

Let $β>0$ and consider an $n$-point process $λ_1, λ_2,..., λ_n$ from Hermite $β$ ensemble on the real line $\mathbb{R}$. Dumitriu and Edelman discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than $\sqrt {\log n}$, i.e., local semicircle law holds. And the number of positive states in $(0,\infty)$ is proved to fluctuate normally around its mean $n/2$ with variance like $\log n/π^2β$. The proofs rely largely on the way invented by Valk$\acute{o}$ and Vir$\acute{a}$g of counting states in any interval and the classical martingale argument.

math.PR↗