# NECESSARY AND SUFFICIENT CONDITIONS FOR THE ASYMPTOTIC DISTRIBUTIONS OF COHERENCE OF ULTRA-HIGH DIMENSIONAL RANDOM MATRICES

@article{Shao2014NECESSARYAS, title={NECESSARY AND SUFFICIENT CONDITIONS FOR THE ASYMPTOTIC DISTRIBUTIONS OF COHERENCE OF ULTRA-HIGH DIMENSIONAL RANDOM MATRICES}, author={Q. Shao and Wen-Xin Zhou}, journal={Annals of Probability}, year={2014}, volume={42}, pages={623-648} }

Let $\mathbf {x}_1,\ldots,\mathbf {x}_n$ be a random sample from a $p$-dimensional population distribution, where $p=p_n\to\infty$ and $\log p=o(n^{\beta})$ for some $0 0$, where $\alpha$ satisfies $\beta=\alpha/(4-\alpha)$. Asymptotic distributions of $L_n$ are also proved under the same sufficient condition. Similar results remain valid for $m$-coherence when the variables of the population are $m$ dependent. The proofs are based on self-normalized moderate deviations, the Stein-Chen method… Expand

#### 17 Citations

Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data

- Mathematics
- 2019

Let $${X}_{k}=(x_{k1}, \ldots , x_{kp})', k=1,\ldots ,n$$ X k = ( x k 1 , … , x kp ) ′ , k = 1 , … , n , be a random sample of size n coming from a p -dimensional population. For a fixed integer… Expand

Asymptotic Analysis for Extreme Eigenvalues of Principal Minors of Random Matrices

- Mathematics
- 2019

Consider a standard white Wishart matrix with parameters $n$ and $p$. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and… Expand

Asymptotically independent U-statistics in high-dimensional testing

- Mathematics
- 2018

Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and… Expand

Distribution-Free Tests of Independence with Applications to Testing More Structures

- Mathematics
- 2014

We consider the problem of testing mutual independence of all entries in a d-dimensional random vector X=(X_1,...,X_d)^T based on n independent observations. For this, we consider two families of… Expand

Largest entries of sample correlation matrices from equi-correlated normal populations

- Mathematics
- 2019

The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the… Expand

Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation

- Mathematics
- 2016

This is an expository paper that reviews recent developments on optimal estimation of structured high-dimensional covariance and precision matrices. Minimax rates of convergence for estimating… Expand

Testing mutual independence in high dimension via distance covariance

- Mathematics
- 2016

We introduce an L2â€ type test for testing mutual independence and banded dependence structure for high dimensional data. The test is constructed on the basis of the pairwise distance covariance and… Expand

Correlation Tests and Linear Spectral Statistics of the Sample Correlation Matrix

- Computer Science, Mathematics
- IEEE Transactions on Information Theory
- 2017

This general CLT can be used to establish the asymptotic behavior of two of the most important correlation test statistics, namely the generalized likelihood ratio test and the Frobenius norm test, under both null and alternative hypotheses. Expand

ARE DISCOVERIES SPURIOUS? DISTRIBUTIONS OF MAXIMUM SPURIOUS CORRELATIONS AND THEIR APPLICATIONS.

- Mathematics, Medicine
- Annals of statistics
- 2018

The multiplier bootstrap procedure is applied to construct the upper confidence limit for the maximum spurious correlation and testing exogeneity of covariates, which provides a baseline for guarding against false discoveries due to data mining and tests whether the fundamental assumptions for high-dimensional model selection are statistically valid. Expand

Distribution-free tests of independence in high dimensions

- Mathematics, Medicine
- Biometrika
- 2017

Two families of distribution-free test statistics, which include Kendall's tau and Spearman’s rho, are studied and it is shown that the two tests are rate-optimal in terms of power against sparse alternatives and that they outperform competitors in simulations. Expand

#### References

SHOWING 1-10 OF 18 REFERENCES

THE ASYMPTOTIC DISTRIBUTION AND BERRY-ESSEEN BOUND OF A NEW TEST FOR INDEPENDENCE IN HIGH DIMENSION WITH AN APPLICATION TO STOCHASTIC OPTIMIZATION

- Mathematics
- 2008

Let $\mathbf{X}_1,...,\mathbf{X}_n$ be a random sample from a $p$-dimensional population distribution. Assume that $c_1n^{\alpha}\leq p\leq c_2n^{\alpha}$ for some positive constants $c_1,c_2$ and… Expand

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

- Mathematics
- 2010

Let {Xk,i; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {pn; n ≥ 1} be a sequence of positive integers such that n/pn is bounded away from 0 and ∞. In this paper… Expand

Some strong limit theorems for the largest entries of sample correlation matrices

- Mathematics
- 2006

Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from 0 and $\infty$. For… Expand

The asymptotic distributions of the largest entries of sample correlation matrices

- Mathematics
- 2004

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… Expand

On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix

- Computer Science, Mathematics
- J. Multivar. Anal.
- 2012

The asymptotic distribution of the largest entry L"n=max"1"@?"i" ~n^[email protected]!"("n"l"o"g"n") is established and six interesting new lemmas which may be of independent interest are presented. Expand

Self-normalized large deviations

- Mathematics
- 1997

Let {X, X n , n ≥ 1} be a sequence of independent and identically distributed random variables. The classical Cramer-Chernoff large deviation states that lim n→ ∞ n -1 ln P((Σ i n =1 X i )/n ≥ x) =… Expand

Asymptotic distribution of the largest off-diagonal entry of correlation matrices

- Mathematics
- 2007

Suppose that we have observations from a -dimensional population. We are interested in testing that the variates of the population are independent under the situation where goes to infinity as . A… Expand

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

- Computer Science, Mathematics
- J. Multivar. Anal.
- 2012

The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension p grows as a function of n. Expand

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

- Mathematics
- 2011

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… Expand

Self-normalized Cramér-type large deviations for independent random variables

- Mathematics
- 2003

Let X 1 , X 2 ,... be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramer-type large deviation… Expand