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Statistics of spectral properties of matrix-valued random variables.

2 votes
Accepted

Error bound for MonteCarlo estimate of elements in Gram-Matrix

$\newcommand{\Om}{\Omega}\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}\newcommand{\De}{\Delta}$The question makes sense only if $A^{-1}$ exists, which will be assumed in what follows. For $x\in\O …
Iosif Pinelis's user avatar
1 vote
Accepted

Expectation of supremum of sub gaussians

$\newcommand\si\sigma$The inequality stated in that lemma is, not $$E\max_{1\le i\le n}X_i \le \max_{1\le i\le n}\si_i^\star\sqrt{\ln(i+1)}, \tag{0}\label{0}$$ but $$E\max_{1\le i\le n}X_i \lesssim \m …
Iosif Pinelis's user avatar
2 votes
Accepted

Orthogonal projection $X X^+$ from random Gaussian matrix $X$

$\newcommand\R{\Bbb R}\newcommand\Si{\Sigma}$The answer to your both questions is yes, regardless of what the covariance matrix $\Sigma$ is. Indeed, let $P_X:=X(X^\top X)^{-1}X^\top$, the orthoproject …
Iosif Pinelis's user avatar
1 vote
Accepted

Distribution of the constraint matrix conditioned on the solution of the linear system

(It will be assumed here that $A$ and $b$ are independent. Of course, without an assumption on the joint distribution of $A$ and $b$, hardly anything can be said about the joint distribution of $A$ an …
Iosif Pinelis's user avatar
2 votes

Distribution of the constraint matrix conditioned on the solution of the linear system

(It will be assumed here that $A$ and $b$ are independent. Of course, without an assumption on the joint distribution of $A$ and $b$, hardly anything can be said about the joint distribution of $A$ an …
Iosif Pinelis's user avatar
1 vote

Is there a closed-form solution for $\max_D \operatorname{Tr}(ADBD)$

For any of your diagonal matrices $D$, let $J:=J_D$ be the set such that $D_{i,j}=1(i=j\in J)$ for all $i,j$ in the set $[N]:=\{1,\dots,N\}$, where, for any matrix $M$, its $(i,j)$-entry is denoted by …
Iosif Pinelis's user avatar
1 vote
Accepted

Estimation on rotationally-disturbed random vectors

$\newcommand\ep\varepsilon$This is impossible to do even for $d=1$, as your model is not identifiable, that is, the parameters of the model are not identifiable even if the distribution of the $X_i$'s …
Iosif Pinelis's user avatar
1 vote
Accepted

Can we show that for every $\delta>0$, there exist constants $\alpha>0, \beta>0$ so that the...

$\newcommand\al\alpha\newcommand\be\beta\newcommand\ep\epsilon\newcommand\de\delta$Let $\al=\beta=1$. Let $Z_n:=\sqrt n\,X_n$, so that $Z_n\to Z\sim N(0,1)$ in distribution. Then for real $\ep>0$ $$ P …
Iosif Pinelis's user avatar
2 votes
Accepted

Bound for expectation of random matrix

No, of course not. Indeed, consider a simplest case when $m=n=1$, and $X:=\mathbf X$ and $y:=\mathbf y$ are iid standard normal random variables. Then $$E(\mathbf{X}^{\mathrm{H}} \mathbf{X})^{-1} \mat …
Iosif Pinelis's user avatar
0 votes
Accepted

Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy...

$\newcommand\la\lambda$The answer is: yes, of course. Indeed, let $X_{N,i}:=N^{2/3}(\la_i-2)$. By the limit theorem you cited and (say) Example 2.3, p. 18, the $k$-tuple $(X_{N,N},\dots,X_{N,N-k+1})$ …
Iosif Pinelis's user avatar
2 votes
Accepted

Distribution of scaled Johnson-Lindenstrauss transforms

$\newcommand\ep\epsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$We have \begin{equation*} P((1-\ep)\|x\|\le\|Ax\|\le(1+\ep)\|x\|)\ge\de \tag{1}\label{1} \end{equation*} for some $\ep,\de …
Iosif Pinelis's user avatar
3 votes
Accepted

Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$...

$\newcommand{\tr}{\operatorname{tr}}$ Let $x=[x^1,\dots,x^p]^T:=x_1$, $y:=xx^T$, $\mu:=Ey$, $w:=y-\mu$, and $s:=\sum_1^n x_ix_i^T$. Then, by the appropriate laws of large numbers, $s/n\to Ey$ almost s …
Michael Hardy's user avatar
2 votes
Accepted

Moments of rescaled Bernoulli random matrix

It is apparently assumed that the $Z_{ij}$'s are independent, as we will do here -- since otherwise hardly anything can be said. Suppose also that $m\ge2$ and $0<p<1$. The $ab$-entry of the matrix $Y: …
Iosif Pinelis's user avatar
3 votes

Expected value of orthogonal projection $X^{+}X$

Assume that $\det\Sigma\ne0$. Then the random matrix $X$ is of rank $m$ almost surely (a.s.). So, a.s. the Moore--Penrose inverse of $X$ is $X^+=X^\top(XX^\top)^{-1}$ and hence $$X^+X=X^\top(XX^\top)^ …
Iosif Pinelis's user avatar
0 votes

Concentration of the norm of subGaussian random vectors

Your desired conclusion holds, even without the additional assumption that $\frac1{\sqrt n}(|y|^2-E|y|^2)$ is a sub-exponential random variable with a sub-exponential norm not dependent on $n$. We onl …
Iosif Pinelis's user avatar

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