5
votes
1answer
92 views

What background is required to understand Random Matrix Theory

I would like to be able to understand RMT but after "reading" many articles I have found that I have a limited capacity. I just see complicated equations. I guess I know linear algebra, classical ...
1
vote
0answers
43 views

Products of Random Matrices

I'm interested in the following process on the space of $d \times d$ real valued matrices, $M_d(\mathbb{R})$. Fix $n \in \mathbb{N}$ and consider the process $$X_{k,n} = \left( I + ...
4
votes
0answers
224 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
1
vote
1answer
64 views

Random operators

Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ...
1
vote
1answer
75 views

Density of Gaussian Unitary Ensemble

I'm trying to learn a bit about Gaussian matrix ensembles, and am having some trouble making the following connection. Sorry if I'm being a bit obtuse. Take the Gaussian unitary ensemble (GUE) of $n ...
0
votes
0answers
10 views

Existence of invers of a function from covariance matrices space

Let $(\mathbb{R}^k,{\cal A})$ be a measurable space. Fix $c>0$ and for every $X\in \mathbb{R}^k$ define $X_c$ as $k-$dimensional vector such that the $i-th$ element of $X^{(c)}$ is $(-c)\vee ...
1
vote
1answer
204 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
0
votes
1answer
97 views

If a sequence of random matrices converge in probability, do their elements also converge?

Is it true that if a sequence of random matrices $\{X_n\}$ converge in probability to a random matrix $X_n\overset{P}{\to}X$ as $n\to\infty$ that the elements $X_n^{(i,j)}\overset{P}{\to} X^{(i,j)}$ ...
2
votes
1answer
46 views

Question regard the notion of almost sure convergence

Consider an $n\times m$ matrix with i.i.d. entries each having zero mean and variance $1/n$. Let $Y = X^TX$. By the strong law of large numbers, we know that the $(i,j)$ entry of $Y$ goes almost ...
2
votes
1answer
70 views

Random Matrix Question

Let $A$ be a $n\times m$ ($n<m$) random matrix with normal i.i.d. entries ($N(0,1)$). Using the law of large numbers, it readily can be shown that as $n\to\infty$ $$ \frac{1}{n}A^TA\to I_m $$ ...
3
votes
0answers
72 views

trace class norms of random matrices

We denote by $||.||_1$ the trace class norm. on $M_n$.Let $(r_{ij})_{1 \leq i,j\leq n}$ be a family of independent identically distributed random variables which take the values $-1$ and $1$ with ...
1
vote
1answer
369 views

Expected value of a multivariate distribution

Given this random vector: $$ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $$ And this probability distribution function that takes it as argument: $$ f_\mathbf{X}(\mathbf{x}) = ...
2
votes
1answer
118 views

Is Wishart Matrix?

Analyzing a system, I have faced a problem which is related to Random Matrices and in particular Wishart matrix. The problem is as follows: Lets assume $\boldsymbol{H}$ is an $m\times n$ random ...
1
vote
0answers
56 views

How to determine the degrees of freedom of an inner-product matrix of two random matrices?

I have two random matrices A and B, with N columns each. The columns of A and B are independent but not necessarily identically distributed. A and B may be considered as two instances of an underlying ...
3
votes
0answers
65 views

Suboptimal confidence for bounds on the extreme eigenvalues of random matrices: can they be improved?

Typical results from the literature of Random Matrix Theory (RMT) (e.g., Tropp'11, Vershynin'12) provide probabilistic guarantees for large deviation of the extreme eigenvalues of random matrices like ...
2
votes
0answers
274 views

Slutsky's theorem for random matrices

This image is from Applied Multivariate Analysis. In this image plim means convergence in probability. I could not find the reference about the statement for ...
11
votes
1answer
482 views

spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d entries with variance $\sigma^2/n$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$ ...
1
vote
0answers
82 views

Studying the maxima of columns of a random matrix as a point process

Consider a matrix, $S$, of i.i.d. real RVs : $X_{ij}$ for $1 \leq i \leq s$, $1 \leq j \leq n$. Let $F$ denote the distribution of $X_{ij}$. For $1 \leq j \leq n$, consider $Y_{j}^{(1)} = \max_{i} ...
5
votes
1answer
149 views

Is there a simple argument for why a random symmetric matrix has distinct eigenvalues?

Lets generate a random symmetric matrix $A$ by generating the entries of a random matrix $Z$ iid from some continuous distribution, and setting $A=(1/2)(Z+Z^T)$. I think its true that $A$ should have ...