# Questions tagged [measure-concentration]

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### Bounds tighter than the additive Chernoff

Additive Chernoff Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$. \begin{gather*} \operatorname{Pr}\left(\...
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### Limiting value of expectation of trace of truncated Gram matrix

Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
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### A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
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Let $(X_i)_{i=1}^N$ be mean zero sub-Gaussian random vectors in $\mathbb{R}^n$, i.e., there exists $C>0$ such that for all $u\in \mathbb{R}^n$, $$\mathbb{E}\left[e^{u^\top X_i}\right]\le e^{\frac{... • 483 2 votes 0 answers 107 views ### Large deviation principle for product of iid bounded symmetric random variables Let n and k be positive integers. Let X be the empirical mean of n iid Rademacher random variables. Note that the distribution of X is symmetric about 0, and also |X| \le 1 w.p 1. Let X_1,... • 6,726 1 vote 1 answer 190 views ### Concentration of a certain simple / well-structured random multilinear polynomial with growing degree Let k and N_1 be positive integers and set N=kN_1. Partition [N] := \{1,2,\ldots,N\} k disjoint from G_1,\ldots,G_k of each of size N_1, and let \mathcal T(k,N_1) be a transversal of ... • 6,726 2 votes 1 answer 305 views ### Converse of the Herbst argument? Background For a real-valued random variable X, define its entropy by H(X) = E[\phi(X)] - \phi(E[X]), where \phi(u) = u \log u. It can be shown that, if the entropy satisfies the bound$$ H(e^{\...
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Suppose $(Z_1, Z_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want ...