# Tagged Questions

0answers
18 views

### the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...
1answer
209 views

### Probability distribution of uAv…

Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix. What is the distribution of $u^HAv$ ( or $||u^HAv||^2$) where : u is a column vector of U. v ...
1answer
137 views

### Distribution of entries of a doubly-sorted random matrix

Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an ...
1answer
182 views

### Concentration of sum of powers of normals

Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables ...
1answer
57 views

### Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix

Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This ...
1answer
120 views

### concentration of random matrices involving normal random variables

Define the random variable \begin{align*} A=|a_1|^2\mathbf{a}\mathbf{a}^* \end{align*} where $\mathbf{a}\in\mathbb{c}^n$ is a random vector distributed as ...
1answer
143 views

### Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...
0answers
68 views

### Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
1answer
118 views

### concentration of sums of fourth moment of normals

I was wondering what is the best tail bound for \begin{equation*} \mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ? \end{equation*} where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.
1answer
215 views

### Measure concentration for weakly dependent random variables

For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a ...
1answer
317 views

### Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $\mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I)$? The kronecker product is where the confusion ...
0answers
674 views

### Distribution of Inverse of a Random Matrix

Recently i got stuck into a problem and couldn't find its satisfactory answer anywhere. My question is simple. Suppose i have a fat random matrix (i,e $R$ has dimensions $k\times d$ where $k<d$) ...
1answer
797 views

### Eigenvalue distributions of finite dimensional Wishart matrices

I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
2answers
721 views

### Does the Tracy-Widom distribution describe the tails of eigenvalue densities of finite dimensional random matrices?

The Tracy-Widom distribution (TW) describes the density of the largest eigenvalue of a random Hermitian matrix, when scaled and centered appropriately (depending on GOE/GUE/GSE/Wishart, etc). In a ...
1answer
1k views

### When should we expect Tracy-Widom ?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
1answer
582 views

### Uniform correlation matrix sampling and not so uniform laws

Hi everyone, I am looking for a way of simulating correlation matrices of fixed dimension in (at least) two ways. First, I would like to determine the "uniform" distribution over the "correlation ...
3answers
726 views

### Marginal distribution of the diagonal of an inverse Wishart distributed matrix

This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here: Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the ...
1answer
495 views

### What are the origin and applications of this result?

In a course taught by Morris Eaton on multivariate statistics that dealt mostly with the Wishart distribution, I learned this proposition: Suppose  M = \begin{bmatrix} A & B \\\\ B^T & C ...