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We define the coefficients $c_{k,k-i}$ of ${n \choose k}$ by the following easy expansion: \begin{align*} & {n \choose k} = \frac{1}{k!} n(n-1) \dots (n-k+1) = \frac{1}{k!} \prod\limits_{t=0}^ …

Is there an easily definable projection $F$ from the orthogonal group $O(n)$ into the permutation group $S_n$, which has the following properties? $F(P_\sigma) = \sigma$ for all $\sigma \in S_n$ $F^ …

Let $U(n)$ be the compact manifold of unitary $(n \times n)$-matrices and let $\mu_n$ denote the Haar-probability measure on $U(n)$. For $m < n$ does there exists a measurable (maybe even continuous o …

While playing around with random matrices and I arrived at a different formula for the mean of the limiting normal distribution for a spectral CLT for sample covariance matrices. More precisely I have …

Let $\mathbb{X}_{p,n}$ be a $(p \times n)$ random matrix whose entries are iid complex standard normal random variables. The hermitian random matrix $\mathbb{S}_{p,n} = \frac{1}{n} \mathbb{X}_{p,n} \m …

First assume $\Sigma = \operatorname{Id}$. Write $A=((a_j)_{i})_{i \leq k, j \leq n}$, then $L_1 := AA^T$ is said to be a real Wishart matrix or $1$-Laguerre matrix. In this case some extremely sharp …

Let $X$ be a random $(p \times n)$-matrix with iid centered columns and suppose the entries of $X$ all have light tails (in a strong enough sense, for example sub-Gaussian). Are there any results boun …

Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\mathb …