Search Results

Let $X_1, X_2$ be two iid random row vectors in $\mathbb{R}^p$, each of whose components are real valued. I'd like to prove that the $\mathbb{R}^{p\times p}$ random matrices $X_1X_1', X_2X_2'$ are al …

Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics …

For each $ m \ge 1$, let $X_m$ and $Y_m$ be two non-negative iid random variables with the same distribution. (The distributions of $X_m$ may change with different $m$.) **Assume that their support of …

I tried posting this question on Cross Validated (the stack exchange for statistics) but didn't get an answer, so posting here: Let's consider a supervised learning problem where $\{(x_1,y_1) \dots (x …

I'm looking for results where perturbation by iid random entries to a matrix will result in convergence of the eigenvalues to the original eigenvalues. More precisely, Let $ \forall n \in \mathbb{N} …

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 happe …

I'm interested to see a result where for large degree of freedom $m,$ the chi distribution $\chi_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $ …

Let $\lfloor{*}\rfloor$ denotes the nearest integer $\le *$. I'm asking myself the question what's the limit of the part of the empirical spectral distribution corresponding to the first $\lfloor{p/2 …

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 entrie …

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 entrie …

Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here …

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 $ …

I asked this on MSE, but got no answer, hence asking here now. Help appreciated! My question is motivated by this question and this question, where the first was aimed for giving a one sided tail bou …

Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X| …

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 …