All Questions
13 questions
1
vote
0
answers
80
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Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
- 193
6
votes
1
answer
1k
views
Largest eigenvalues of a (random) correlation matrix?
I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
- 380
2
votes
1
answer
137
views
Local distribution of sample covariance matrix when the number of observations/realisations is less than the matrix dimension
Given a true covariance matrix $M$ of dimension $p \times p$, we generate $n$ gaussian random vectors $X_1,..X_n \sim N(0,M)$. We then get a sample covariance matrix $M_s$ based on these $n$ ...
- 159
3
votes
2
answers
581
views
Largest eigenvalue of the adjacency matrix of weighted random graph
I find the theorem for largest eigenvalue of the adjacency matrix of ER random graph in here https://arxiv.org/pdf/math/0106066.pdf. The adjacency matrix is a symmetric random matrix s.t. diagonal ...
- 272
3
votes
0
answers
151
views
Largest eigenvalue divided by $n$
Let $X$ be an $n\times n$ symmetric random matrix whose diagonal is fixed as $1$, and every element in the upper triangle (excluding the diagonal) is drawn from Bernoulli($p$). The elements in the ...
- 272
3
votes
0
answers
436
views
Rank of Hadamard product with random matrices
I do research in statistics and am not sure whether the following is considered research level or not in mathematics. If it isn't, I'm happy because that means the answer is probably known and I can ...
- 131
4
votes
0
answers
188
views
Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
1
vote
0
answers
201
views
Rank of cross-covariance matrix
Let $\boldsymbol{X}=(X_1,\dots,X_p)^T$ and $\boldsymbol{Y}=(Y_1,\dots,Y_q)^T$ be two random vectors. Denote $r_x=\text{rank}(\text{Cov}(\boldsymbol{X})),r_y=\text{rank}(\text{Cov}(\boldsymbol{Y})), r_{...
- 193
4
votes
1
answer
294
views
Finding high-dimensional correlation matrices that are both sparse and low-rank
Let $\boldsymbol{R}$ be the correlation matrix of $X_i,i=1,\dots,p$ with a large $p\gg q=\text{rank}(\boldsymbol{R})$. Is that reasonable to assume that $\boldsymbol{R}$ is both (approximately) sparse ...
- 193
2
votes
0
answers
366
views
Convergence rate of Pearson correlation matrix
I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
- 121
2
votes
2
answers
1k
views
Gaussian expectation of an exponentiated outer product
Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation,
$$ E\left[ \exp(\mathbf{xx}^\top)\right]$$
where $\exp(\cdot)$ is element-wise exponential function (not ...
- 291
4
votes
1
answer
189
views
Weak ergodicity of nonhomogenous products of 0-1 matrices
Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...
- 4,679
2
votes
2
answers
739
views
Multinomial transformation for matrices
Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way:
$r_i=\log(...
- 203