All Questions
Tagged with stochastic-processes random-matrices
25 questions
2
votes
0
answers
116
views
Random matrix with power law decay in eigenvalues
What positive semi-definite random matrices have (roughly) $n^{-\alpha}$ for $n^{th}$ singular value? The power law decay need not be exact.
I want to find random matrix ensembles that naturally ...
- 433
3
votes
1
answer
307
views
Request for references of random matrices
I need some good books aimed as a detailed and gentle introduction to random matrices, containing good discussion and derivation of Marchenko–Pastur distribution. Also, I request some other references ...
- 826
1
vote
1
answer
257
views
Using gradient descent in probability case
Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of:
$$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i,...
- 421
4
votes
1
answer
346
views
Mehta integral and orthogonality
The Mehta integral is the following expression:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-t_i^2/2}
\prod_{1 \le i < j \le n} |t_i - t_j |^{2 \...
- 124
3
votes
1
answer
373
views
Matrix positive semi-definite
We construct a non-random matrix using random variables as follows:
We fix the vector $v=(1,1).$
Let $X$ be a $\mathbb R^2$-valued random variable such that $X$ is distributed according to
$$d\mu(...
- 192
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
3
votes
2
answers
307
views
Random matrix is positive
This is a follow up question on my previous question here that was on solved in the deterministic setting by Denis Serre, when the perturbation can be separated. Therefore, I decided to split the ...
- 536
3
votes
0
answers
184
views
Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion
I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...
- 41
-2
votes
1
answer
963
views
On the Cauchy-Schwarz Inequality for trace function of random matrices
In the deterministic case, for two matrices $A$ and $B$ with appropriate matrices, we know that
$$tr((A^{T}B)^{2})\leq tr(A^{T}A)tr(B^{T}B)$$
which is the trace form of Cauchy-Schwarz-Inequality (CSI)....
- 77
1
vote
1
answer
287
views
Random matrix and spherical spin-glass
The Hamiltonian of the p-spherical spin glass model is
$$H_{N,p}(\sigma)=\frac{1}{N^{\frac{p-1}{2}}} \sum_{i_1,...,i_p=1}^N X_{i_1,...,i_p} \sigma_{i_1}\cdot...\cdot \sigma_{i_p}$$
where $\sigma \in ...
- 536
1
vote
0
answers
104
views
Trivial zeros of $\zeta$ from limit characteristic functions of random matrices
Reviewing some of the literature on random matrices I have seen several studies and results on characteristic polynomials of random matrices, usually of fixed size/degree $N$. Zeros then are either on ...
- 982
2
votes
0
answers
123
views
Modified Wigner semicircle law
The Wigner semicircle law states that for a random GOE-matrix $M^N \in \mathbb{R}^{N \times N}$ in the $N \rightarrow \infty$ limit for any $f \in C^b(\mathbb{R})$
$$\lim_{N \rightarrow \infty}\frac{...
user118408
1
vote
0
answers
146
views
minimum eigenvalue of Katri-Rao product of two Gaussian matrices
Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum ...
- 363
1
vote
0
answers
295
views
One-sided Talagrand concentration inequality for empirical processes
Let $\mathcal{F}$ denote a function class. A classic result by Talagrand states that
\begin{align*}
\mathbb{P}\bigg\{\sup_{f\in\mathcal{F}}\big|\sum_{i=1}^nf(X_i)-\mathbb{E}\big[\sum_{i=1}^nf(X_i)\...
- 859
4
votes
0
answers
416
views
concentration of functions of Gaussian processes
Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
- 859
2
votes
0
answers
54
views
Construction of point process having same pair correlations as GUE
The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately)
$$
g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + \delta(u)....
- 2,207
11
votes
8
answers
2k
views
Semicircle law universality elsewhere
Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
- 4,952
3
votes
0
answers
82
views
Matroid rank decay
Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field $...
- 161
2
votes
2
answers
219
views
Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)
Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have
$$X_n = A_n \cdots A_1X_0.$...
- 155
2
votes
0
answers
1k
views
Random matrices whose limit gives exact Wigner surmise
Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write $\rho^W_0(s)=\...
- 4,952
1
vote
1
answer
257
views
maximum of certain Gaussian processes
Let $\mathbf{a}_k\in\mathbb{C}^n$ for $k=1,2,\ldots,m$ be i.i.d. standard complex normal random vectors with distribution $c\mathcal{N}(0,\mathbf{I})$. I am interested in a tight upper bound on the ...
- 859
4
votes
1
answer
637
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 ...
- 4,952
2
votes
1
answer
2k
views
Bounds on the eigenvalues of a random binary matrix
Consider $A$, a random binary matrix of zeros and ones in $\mathbb{R}^{{M\times N}}$, and $M>N$. We assume that $P(a_{i,j}=0)=P(a_{i,j}=1)=0.5$ (although I appreciate any advice on the case of non-...
- 127
1
vote
0
answers
132
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 ...
- 11
0
votes
1
answer
554
views
Stochastic processes with random matrices
I am currently working on complex networks. I consider a matrix $\cal N$ with random entries $\delta_{ik}$. These entries are varying randomly in time and so I have a sequence of random matrices that ...
- 1,687