All Questions
Tagged with free-probability pr.probability
46 questions
2
votes
0
answers
56
views
Sum of independent Wisharts
Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
- 193
1
vote
1
answer
84
views
Limiting value of Stieltjes transform of sum of independent Wishart matrices
Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
- 6,853
0
votes
1
answer
108
views
RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)
Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
- 6,853
0
votes
0
answers
42
views
Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
- 6,853
1
vote
2
answers
301
views
Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$
Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
...
- 2,442
1
vote
0
answers
76
views
Moment method / genus expansion for random matrices with i.i.d. entries
Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$
...
- 31
8
votes
3
answers
506
views
Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
- 484
2
votes
0
answers
119
views
Random matrices may be asymptotically free but never free themselves?
It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
- 21
19
votes
0
answers
3k
views
What does a product of many Gaussian matrices converge to?
Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$.
Is ...
- 2,442
2
votes
2
answers
215
views
How to analyze the value of convergence of functions of random matrices?
Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
- 287
2
votes
0
answers
96
views
Limiting value of $\dfrac{1}{m}\mathrm{tr}(FAF^\top (FBF^\top)^{-1})$, where $F$ has iide $N(0,1)$ entries and $A,B$ are deterministic
Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
- 6,853
1
vote
0
answers
68
views
Limiting value of expectation of $\operatorname{tr}(BR(z))$, where $R(z) := (X^\top X - z I_d)^{-1}$ and $X \sim N_{n,d}(0,A)$
Let $A=A(d)$, and $B=B(d)$ be (sequences of) deterministic positive-definite $d \times d$ matrices and let $X$ be an $n \times d$ random matrix with iid rows from $N(0,A)$. Let $R$ be the resolvent of ...
- 6,853
9
votes
1
answer
371
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 ...
4
votes
1
answer
164
views
Limiting value of expectation of trace of exponential of Wishart matrix
Let $X$ be an $n \times d$ random matrix with iid entries from $N(0, 1/d)$. Let $S:=X^\top X/n$, a $d \times d$ Wishart matrix and let $T = e^{S} := \sum_{k=0}^\infty \dfrac{S^k}{k!}$ be its ...
- 6,853
3
votes
0
answers
93
views
Explaning why the spectrum of a setting simple structure random matrix is always spiked ($d-1$ eigenvalues close to zero, and $1$ away from zero)
For concreteness, let $m=500$, $d=600$, $N=1000$. Let $W$ be and $d \times m$ matrix with unit-norm rows and let $u$ be a uni-norm vector of length $m$. Given a binary vector $b$ of length $m$, length ...
- 6,853
5
votes
0
answers
161
views
Determinant bounded away from zero
Suppose $P(A_1,\dots,A_k,A_1^{-1},\dots,A_k^{-1})$ is a noncommutative polynomial with positive coefficients. We may then consider the map $g:U(n)^k\rightarrow\mathbb{C}$ from the unitary group $U(n)$,...
- 163
0
votes
0
answers
91
views
Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function
Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
- 6,853
1
vote
1
answer
104
views
Limiting value of $\dfrac{1_n^\top B^{-1} A B^{-1} 1_n}{d}$, where $A=WW^\top + a I_n$, $B = WW^\top + b I_n$, and $W \sim N(0,\Sigma_d)$
Let $n$ and $d$ be positive integers with
$$
n,d \to \infty, \quad n/d \to \rho \in (0,\infty).
$$
Let $\Sigma_d$ be a psd matrix such that
$\mbox{trace}(\Sigma_d) = 1$.
$\|\Sigma_d\|_{op} = \mathcal ...
- 6,853
20
votes
2
answers
2k
views
Is there a noncommutative Gaussian?
In classical probability theory, the (multivariate) Gaussian is in some sense the "nicest quadratic" random variable, i.e. with second moment a specified positive-definite matrix. I do not ...
- 5,701
0
votes
1
answer
153
views
Q-Gaussian processes and probability space
It is well known that for isonormal Gaussian processes, one can decompose the space $L^2(\Omega)$ into Wiener chaos so that the space $L^2(\Omega)$ is isomorphic to the direct sum of the Wiener chaos, ...
- 311
1
vote
1
answer
126
views
Probabilistic lower and upper-bounds for a certain random quartic form involving gaussian random matrices
Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the ...
- 6,853
2
votes
0
answers
172
views
Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix
Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...
- 6,853
3
votes
1
answer
176
views
Combinatorial formula to compute the moments of the product of two free random variables
I found in the PhD thesis Moments method for random matrices with applications to wireless communication the following combinatorial formula to compute the free moments of the product of two random ...
- 77
1
vote
1
answer
295
views
How to compute the first moment of the distribution of the convolution of Marcenko-Pastur law with a not iid matrix?
Let $\mathbf{F}$ denote an M × N matrix whose entries are independent zero-mean complex random variables, the limiting eigenvalue distribution is given by the Marchenko Pastur law $MP_{\beta}$, where $...
- 77
1
vote
1
answer
128
views
Limiting eigenvalue distribution of $YY^\top$ where $Y_{ij} = X_{ij} + a$ and $X$ has iid rows from an isotropic log-concave distribution
Let $a \in \mathbb R$ be a determinstic scalar and let $X$ be and $n \times d$ such that the $n \times n$ psd random matrix $S=XX^T$ has limiting eigenvalue distribution $\mu$, when $n,d \to \infty$ ...
- 6,853
2
votes
1
answer
212
views
For fixed $\lambda \ge 0$, Integrate the function $f_\lambda(x):=x/(x + \lambda)^2$ w.r.t. Marchenko-Pastur density
In trying to solve another the problem posed in the question https://www.mathoverflow.net/q/385777/78539, I'm led to consider the following problem.
Let $\mu_\gamma$ be the Marchenko-Pastur ...
- 6,853
2
votes
1
answer
415
views
High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)
Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
- 6,853
3
votes
0
answers
244
views
Using linearization trick (free probability) to compute limiting singular-value density of $R=XY+Z+A$ (or equivalently, of $RR^\top$)
Disclaimer. I only started learning the subject of free probability $1$ day ago, and I'm still trying to absorb the fundamentals, while applying them to my own specific problems arizing in the ...
- 6,853
4
votes
3
answers
800
views
Quick derivation of classical probability theory from von Neumann algebraic framework
Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko https://www.youtube.com/watch?v=F8Urtr39jM0, I'm led to consider the following question
Question. How can one ...
- 6,853
1
vote
1
answer
144
views
Bounds for the extreme singular-values of random matrix with thresholded entries
Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...
- 6,853
2
votes
1
answer
1k
views
Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data
Let $Z=[z_1, \dots z_n]$ be a $d \times n$ matrix, where the $z_i$'s are iid random vactors with mean $\mu \in \mathbb{R}^d$ and $d \times d$ (population) covariance matrix $\Sigma$, but the entries $...
- 1,465
2
votes
1
answer
210
views
Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the "features" are correlated: references
Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
- 1,465
0
votes
0
answers
115
views
Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?
Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
- 1,465
0
votes
1
answer
320
views
Marcenko Pastur law when the dimensionality/sample size ratio $p/n \to 0, \infty$? Lack of resources?
Let $X: \Omega \to \mathbb{R}^{p \times n}$ be a random matrix so that each entry $X_{ij}$ is a random variable with $\mathbb{E}X_{ij}=0, \mathbb{E}X_{ij}^2=\sigma^2$
I was wondering what would ...
- 1,465
0
votes
1
answer
307
views
Berry-Esseen type theorem for Monotonic independence
The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases ...
- 152
1
vote
0
answers
200
views
Convergence in probability in the setting of free probability
Let $A_n$ and $B_n$ be sequences of positive-definite random matrices whose empirical spectral distributions converge to (possibly different) limiting spectral distributions $\mathcal A$ and $\mathcal ...
- 321
5
votes
3
answers
998
views
Why a random variable is better described by its cumulants than by its characteristic function?
It is a classical and well known problem that a random variable $X$ is not uniquely determined by its moments $\mathbb{E}(X_n)$. The moment problem is the problem of determining the probability ...
- 137
27
votes
3
answers
4k
views
Why is free probability a generalization of probability theory?
Note: This question was already asked on Math.SE nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope ...
- 2,680
3
votes
1
answer
145
views
Existence of free operators, independent and with given distributions
Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...
- 421
4
votes
1
answer
279
views
Spacing of the largest singular values of Wishart matrix
Let $X \in \mathbb{R}^{n \times p}$ consist of iid $\mathcal{N}(0,1)$. Assume that $n/p$ converges to a positive constant. Denote by $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_{\min(n,p)} \ge 0$ the ...
- 773
4
votes
1
answer
813
views
Distribution of sum of freely independent Marchenko-Pastur measures
Given freely independent random variables $X_i$ with Marchenko-Pastur measures $\mu_i$, $i\in\{1,\dots,n\}$ how can we find the distribution of the scaled sum of these random variables $\sum_ia_iX_i$....
- 213
10
votes
2
answers
2k
views
Intuition behind the spectral density of random matrices
Hi,
I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the uniform distribution on ...
- 349
43
votes
8
answers
3k
views
How to quantify noncommutativity?
If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
- 1,890
4
votes
1
answer
714
views
Classical convolution VS Free Convolution
We denote $\varphi:\mathbb R^2\rightarrow\mathbb R$ the addition of real numbers, and $\varphi_*:M_1(\mathbb R^2)\rightarrow M_1(\mathbb R)$ the induced push-forward map (where $M_1(\Delta)$ stands ...
- 2,135
4
votes
0
answers
331
views
What is 'arch' in Vershik-Kerov's 1984 paper?
In their 1984 paper Asymptotic of the Largest and the Typical Dimensions of Irreducible Representations of a Symmetric Group, Vershik and Kerov use the notation $\DeclareMathOperator{\arch}{arch}\arch ...
- 445
6
votes
0
answers
1k
views
Relationship between R-transform and free convolution of random matrices?
I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...
- 1,890