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2
votes
0answers
464 views

Concentration of sum of independent random variables

Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$. Then we can ...
5
votes
0answers
202 views

Chernoff bound in the not-quite-sub-exponential case

In Terry Tao's notes on Concentration of measure, Exercise 7 indicates that the Chernoff bound can be generalized to sub-exponential random variables: ...
5
votes
2answers
365 views

Local concentration of measure on Erdos-Rényi graph

Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
3
votes
1answer
358 views

Chernoff-Hoeffding bound for complex values

Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value $\mu$ and satisfying $|X_i| \le b$. Let $\epsilon > 0$. ...
2
votes
1answer
394 views

concentration inequality for averages of dependent random variables

Let $X \in R^n$ be a random vector such that $$P(|X_i| > \epsilon) \le e^{-\epsilon^2}$$ What is a tight bound on $$P(\sum_{i=1}^n |X_i| > \epsilon)$$ and on $$P(\max_{1\le i\le n} |X_i| ...
8
votes
2answers
528 views

Concentration bounds for sums of random variables of permutations

I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds. As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...
2
votes
0answers
218 views

Does Multiplicative Version of Azuma's Inequality Hold?

It is known that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound. Chernoff ...
5
votes
2answers
793 views

Tails of sums of Weibull random variables

Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = ...
4
votes
1answer
452 views

An elementary probability question

Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite. Consider $n+1$ samples $X_0, \ldots, X_n ...
-1
votes
2answers
446 views

Approximating a subspace by sampling a base without replacement

Let $X$ be a $p \times n$ matrix, with $p > n$. Now, suppose I sample $m < n$ columns from $X$ at random, without replacement. I would like to characterize the distance between the subspace ...
1
vote
0answers
206 views

A Multiplicative version of McDiarmid's Inequality like the one of Chernoff-Hoeffding Bounds

McDiarmid's Inequality basically says the following: Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the ...
0
votes
2answers
252 views

Weak convergence of the image of a weakly $L^1$ converging sequence

This is a follow-up on another question. Can something be said about the image of a weakly converging sequence in $L^1$? More precisely $u_k\ge 0$ $\|u_k\|_{L^1}=\int u_k=1$ $u_k$ converges to $u$ ...
2
votes
2answers
448 views

Weak convergence of the image of an $L^1$ converging sequence under a convex function

Suppose that $u_k$ is a sequence of $L^1$ functions defined on a compact $K\subset R^n$ and a function $f:[0, \infty)\to[0, \infty)$ with the following properties $u_k\ge 0$ $\|u_k\|_{L^1}=\int ...
10
votes
1answer
696 views

Non-probabilistic proof of the Johnson–Lindenstrauss lemma

The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...
1
vote
1answer
339 views

Azuma's Inequality when the conditions hold with high probability?

In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
7
votes
0answers
260 views

How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are $m$ independent $d\times d$ random matrices and let $\overline{X} = \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices ...
1
vote
0answers
156 views

When does the effective concentration of measure does not occour on a Riemmanian manifold?

Introduction Let $\mathcal{M}$ be a compact $m$ - dimensional Riemmanian manifold with normalized measure $\mu$ (derived from the metric). It is know that in this setting we have concentration ...
6
votes
1answer
414 views

First nonzero eigenvalue of the Laplacian on the submanifold

Consider a compact, connected $n$ dimensional Riemmanian manifold $\mathcal{N}$ and its $m$ dimensional closed submanifold $\mathcal{M}$ (with the metric coming from from the one defined on ...
3
votes
0answers
212 views

Concentration of functions of random unitary matrices

Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials ...
4
votes
3answers
517 views

Lower bound for Gaussian random vector with negative correlation

Let $X = (X_1,\ldots,X_n) \in \mathbb{R}^n$ be jointly Gaussian with mean $0$, covariance matrix: $Var(X_i) = 1$, $Cov(X_i, X_{i+1}) = -1/2$, and $Cov(X_i, X_j) = 0$ else. Let $\zeta \in ...
7
votes
2answers
381 views

construction of a random measure with a given mean

Let me first pose a trivial question. Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$? The answer is ...
8
votes
2answers
1k views

Levy's isoperimetric inequality for sphere

Let me recall subj: If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in ...