arXiv subjects

Jiang Hu

Publications and source records attributed to Jiang Hu.

111 records · Page 7Linked to original sources

arXiv2013

Convergence of Empirical Spectral Distributions of Large Dimensional Quaternion Sample Covariance Matrices

In this paper we establish the limit of the empirical spectral distribution of quaternion sample covariance matrices. Suppose $\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}$ is a quaternion random matrix. For each $n$, the entries $\{x_{ij}^{(n)}\}$ are independent random quaternion variables with a common mean $μ$ and variance $σ^2>0$. It is shown that the empirical spectral distribution of the quaternion sample covariance matrix $\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*$ converges to the M-P law as $p\to\infty$, $n\to\infty$ and $p/n\to y\in(0,+\infty)$.

math.PR↗

arXiv2013

Strong representation of weak convergence

Skorokhod's representation theorem states that if on a Polish space, there is defined a weakly convergent sequence of probability measures $μ_n\stackrel{w}\toμ_0,$ as $n\to \infty$, then there exist a probability space $(Ω, \mathscr F, P)$ and a sequence of random elements $X_n$ such that $X_n\to X$ almost surely and $X_n$ has the distribution function $μ_n$, $n=0,1,2,\cdots$. In this paper, we shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces $S_n$, a sequence of probability measures $μ_n$ and a sequence of measurable mappings $φ_n$ such that $μ_nφ_n^{-1}\stackrel {w}\toμ_0$, then there exist a probability space $(Ω,\mathscr F,P)$ and $S_n$-valued random elements $X_n$ defined on $Ω$, with distribution $μ_n$ and such that $φ_n(X_n)\to X_0$ almost surely. In addition, we present several applications of our result including some results in random matrix theory, while the original Skorokhod representation theorem is not applicable.

math.PR↗

arXiv2011

A Note on Rate of Convergence in Probability to Semicircular Law

In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$ when the dimension $n$ tends to infinity.

math.PR↗