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Sharp concentration of the chromatic number on random graphs, Eli Shamir and Joel Spencer (1987). The concentration of measure phenomenon is used to prove concentration of the chromatic number for ra …

the variance of $Y$ is smaller than the variance of $X$ by a factor $\sqrt{1-\frac{n-1}{N-1}}$; for a derivation, see for example section 1.2 of these notes.

Concentration Inequalities for Bounded Random Vectors, by Xinjia Chen (2013): We derive simple concentration inequalities for bounded random vectors, which generalize Hoeffding's inequalities fo …

An integral approximation may be useful. Taking a Poisson distribution for $X$ with intensity $\lambda=np$, I approximate $$E_\lambda(X\log X)=\sum_{k=1}^\infty \lambda^k e^{-\lambda}\frac{k\log k}{k! …

Let me calculate the expectation value of $\alpha$. The probability distribution of $X$ is invariant under orthogonal transformations, so without loss of generality I can orient the unit vector $w$ al …

The first two moments of $X_i$ are given, $\mathbb{E}[X_i]=\mu_i$, $\mathbb{E}[X_i^2]=\sigma_i^2+\mu_i^2$. We also need the fourth moment, $\mathbb{E}[X_i^4]=\tau_i^4$. Then with $Y =\sqrt{\sum_{i=1}^ …

For simplicity, I take equal $\mu_i=\mu$, $\sigma_i^2=\sigma$. The generalisation to arbitrary $\mu_i,\sigma_i$ is straightforward, $k(\mu/\sigma)^2\mapsto\sum_{i=1}^k(\mu_i/\sigma_i)^2={\cal O}(k).$ …

The probability distribution of the largest singular value of $A$ (or the largest eigenvalue $\lambda_1$ of $AA^T$) is derived in Distribution of the largest eigenvalue for real Wishart and Gaussian r …

A random PSD matrix $M$ can be constructed by taking $M=WW^T$, with the $n\times n$ matrix elements of $W$ i.i.d. with mean zero and variance $\sigma^2$. For $n\gg 1$ the marginal distribution $\rho(\ …

Let me try to answer the updated question. If $A$ and $B$ are Hermitian $m\times m$ matrices with $B$ of rank $r$ then the two sequences of eigenvalues $\lambda_k(A+B)$ of $A+B$ and $\lambda_k(A)$ of …