All Questions
12 questions
20
votes
6
answers
19k
views
Intuition for Haar measure of random matrix
What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?
My understanding for what Haar measure means for $U(1)$ is that it ...
- 1,890
17
votes
1
answer
9k
views
Intuitive understanding of the Stieltjes transform
I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does.
The gist of my work is that I have an $N\times N$ true covariance ...
9
votes
1
answer
372
views
Why impossible events have some drawbacks or pathologies in probability theory?
It is said by Halmos, P.R.; in "Lectures on ergodic theory"
"Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
3
votes
2
answers
513
views
Sample from a delta-ball in the orthogonal group O(n)
An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...
- 201
3
votes
1
answer
354
views
Determining the asymptotic behavior of random matrices with vanishing ratio dimensions
Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.
There are a lot of results regarding the asymptotic ...
- 165
2
votes
1
answer
560
views
Covariance matrix as optimization problem solution?
I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= \int_{\...
- 5,405
2
votes
1
answer
148
views
Bound the conditional expectation of a random matrix under weak dependence
Let $X$ be an $d\times d$ random matrix satisfying $\mathbb{E}[X]=0$ and $\|X\|_2\leq 1$ almost everywhere. Let $\mathcal{F}$ be the $\sigma$-field generated by $X$. Now suppose we have another $\...
- 323
2
votes
1
answer
145
views
Calculate correlation values of an ensemble of $N\times N$ real asymmetric random matrix from Gaussian measure
I am now reading a paper by Sommers, H. J., et al. "Spectrum of large random asymmetric matrices." Physical Review Letters 60.19 (1988): 1895-1898., it claims a mathematical statement (equation (2) in ...
- 325
0
votes
1
answer
378
views
Concentration of norm of linearly transformed normal random vector as dimension go to infinity
Earlier asked on MSE, but didn't get an answer, so posting here:
Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
- 1,467
0
votes
1
answer
58
views
Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$
Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral ...
- 6,853
0
votes
0
answers
97
views
Wigner semicircle law and random measures
tl;dr: the proof of the Wigner semicircle law seems to confuse measures with random measures. I do not understand why. Scroll down until 'QUESTION' if you are fine with the theoretical stuff.
T. Tao ...
- 91
-1
votes
1
answer
138
views
On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components
Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.
Now, I was wondering ...
- 1,467