Tiefeng Jiang

Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices

Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n× p random matrix in the high-dimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high-dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.

Complete convergence and almost sure convergence of weighted sums of random variables

AbstractLetr>1. For eachn≥1, let {Xnk, −∞<k<∞} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee

How many entries of a typical orthogonal matrix can be approximated by independent normals

\Sigma _{n \geqslant 1} n^{r - 2} P\{ |\Sigma _{k = - \infty }^\infty X_{nk} | \geqslant \varepsilon \}< \infty

SPECTRAL DISTRIBUTIONS OF ADJACENCY AND LAPLACIAN MATRICES OF RANDOM GRAPHS

for every ε>0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods ofiid random variables are also established.

Phase transition in limiting distributions of coherence of high-dimensional random matrices

Let X_n=(x_{ij}) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let R_n=(\rho_{ij}) be the p\times p sample correlation matrix of X_n; that is, the entry \rho_{ij} is the usual Pearsons correlation coefficient between the ith column of X_n and jth column of X_n. For contemporary data both n and p are large. When the population is a multivariate normal we study the test that H_0: the p variates of the population are uncorrelated. A test statistic is chosen as L_n=max_{i\ne j}|\rho_{ij}|. The asymptotic distribution of L_n is derived by using the Chen-Stein Poisson approximation method. Similar results for the non-Gaussian case are also derived.

Circular law and arc law for truncation of random unitary matrix

We solve an open problem of Diaconis that asks what are the largest orders of p n and q n such that Z n , the p n x q n upper left block of a random matrix r n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the variation distance between the joint distribution of entries of Z n and that of p n q n independent standard normals goes to zero provided p n = o(√n) and q n = o(/n). We also show that the above variation distance does not go to zero if p n = [x√n] and q n = [y√n] for any positive numbers x and y. This says that the largest orders of p n and q n are o(n 1/2 ) in the sense of the above approximation. Second, suppose Γ n = (γ ij ) n×n is generated by performing the Gram-Schmidt algorithm on the columns of Y n = (y ij ) n×n , where {y ij ; 1 ≤ i, j ≤ n} are i.i.d. standard normals. We show that e n (m)::= max 1≤i≤n 1≤j≤m √n. γ ij -y ij | goes to zero in probability as long as m = m n = o(n/log n). We also prove that e n (m n ) → 2√α in probability when m n = [nα/logn] for any a > 0. This says that m n = o(n/log n) is the largest order such that the entries of the first m n columns of Γ n can be approximated simultaneously by independent standard normals.

Limit theorems for beta-Jacobi ensembles

University of Minnesota Ph.D. dissertation. December 2011. Major: Statistics. Advisor: Tiefeng Jiang. 1 computer file (PDF); viii, 72 pages.

Large deviations for moving average processes

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that: (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval almost surely; (iii) the empirical distributions of the eigenvalues of the Laplacian matrices converge weakly to the free convolution of the standard Gaussian distribution and the Wigner’s semi-circular law; (iv) the empirical distributions of the eigenvalues of the adjacency matrices converge weakly to the Wigner’s semi-circular law.