Questions tagged [measure-concentration]
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397 questions
2
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366
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Convergence rate of Pearson correlation matrix
I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
5
votes
1
answer
580
views
Concentration of spectral norm
Let $X_{ij}$ be independent (but not identically distributed) real-valued random variables for $1\leq i\leq j\leq n.$ Let $X$ be the symmetric matrix whose entries are given by $X_{ij}.$ Let
\begin{...
1
vote
0
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361
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concentration inequalities for quadratic forms of correlated random vectors
Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ ...
2
votes
0
answers
246
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Tail bounds for suprema of random processes
Classical results concerning concentration of Gaussian random variables due to Cirelson, Ibragimov and Sudakov say that if $V_1,\cdots,V_n$ are jointly Gaussian with variance bounded by $1$, then (...
3
votes
1
answer
339
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Probability of Hamming weight
Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick ${n^t}$ random vectors $\{v_i\}_{i=1}^{{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$.
Denote $v_j\cap v_j$ to be ...
3
votes
0
answers
270
views
Hamming weight probability of projections
Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick $2^{n^t}$ random vectors $\{v_i\}_{i=1}^{2^{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$.
If $v_i^\perp$ is ...
5
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0
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138
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Improving concentration estimates by controlling sums on subsets
Let $X_1, \dots, X_N$ be uniform random variables (r.v.) in $[-1, 1]$, and let $S_N$ be their sum $S_N = \sum_{i=1}^N X_i$.
If the r.v. are taken independent, then the CLT suggests that $S_N$ is ...
3
votes
1
answer
247
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Concentration and Correlation for Magnitudes of Gaussian Vectors
Suppose we have a large collection of standard normal random variables $a_i\in\mathbb{R}^n$. We know by standard concentration results that if we take $m \geq C\left(t/\epsilon\right)^2n$ samples, ...
-1
votes
1
answer
104
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Question about measure lemma?
"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ $|...
3
votes
1
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460
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Derive concentration bound for the derivative
It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian?
In ...
2
votes
0
answers
145
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Concentration bound in high min entropy distribution
Let $(X_{1},\dots,X_{m})$ be joint distribution on $\{0,1\}^{m}$ with that $H_{\infty}(X_{1},\cdots,X_{m})\geq m-r$, where $H_{\infty}$ means min-entropy.
Let $P_{1},...,P_{n}\subseteq [m]$ be sets ...
1
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0
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151
views
Does Newtonian capacity increase strictly when mass is spread?
We start with two disjoint compact sets A and B with positive capacities. Then, we translate B s.t. $B+rv$ is disjoint from A and B and ,more importantly, $dist(x,y)<dist(x,y+rv)$ for all $x\in A$ ...
5
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0
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327
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Eigenvalues of Random Regular Bipartite Graphs
I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
1
vote
1
answer
239
views
Concentration bound for a martingale-like setting (the expected difference decreases as the sequence increases)
I went through several martingales concentration bounds, but none of them fit the settings I am interested in, which is the following. Suppose I have a sequence of nonnegative random variables $0=Y_{0}...
1
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0
answers
112
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Tools to bound the singular values of a finite sum of random matrices from below?
Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
0
votes
1
answer
196
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reverse FKG type inequality for slightly correlated Gaussian vectors
Let $X$ be a $m$-dimensional Gaussian vector, and $Y$ a $q$-dimensional Gaussian vector, for some $m,q\geq 1$. Assume that the $X_i$ and $Y_j$ are centred and have unit variance. Assume that $E X_i ...
5
votes
1
answer
284
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concentration of random matrices involving normal random variables
Define the random variable
\begin{align*}
A=|a_1|^2\mathbf{a}\mathbf{a}^*
\end{align*}
where $\mathbf{a}\in\mathbb{c}^n$ is a random vector distributed as $\mathcal{N}(0,\mathbf{I}/2)+i\mathcal{N}(0,\...
5
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0
answers
137
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Large Deviations: Exponential decay in normed spaces
Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and $S_n=\frac{1}{n}[...
5
votes
1
answer
705
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Expectation of ratio of functions of i.i.d. Bernoullis: a concentration question
Consider the following $n \times n$ symmetric matrix of i.i.d. Bernoulli random variables, $X_{ij}$. For $i=1,...,n$ and $i<j\le n$. Let $X_{ij} \sim \text{Bernoulli}(p)$ when $i \ne j$ ($p$ fixed)...
3
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0
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152
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Bounding expected value of maximum of dot product with random chirp
Let $\mathbf{x}\in\mathbb{C}^n$ with $\|\mathbf{x}\|=1$ with $n<\frac{N}{2}$. I am interested in a bound of the form
\begin{equation*}
\mathbb{E}\Big\{\max_{k\in\{1,2,\ldots,n\}}\Big|\sum_{a=1}^ne^{...
5
votes
0
answers
117
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Concentration of weighted random chirp
I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have
\begin{align*}
\mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}...
10
votes
1
answer
2k
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Bounds on the moments of the binomial distribution
I'm looking for simple and reasonably tight bounds on the k-th moment of the Binomial distribution $B(n,p)$, namely, $E[B(n,p)^k]$. I'm interested in the case when k is large (say on the order of $\...
4
votes
1
answer
474
views
Concentration inequalities in $\ell_{\infty}$ for sums of iid random ("nice") functions?
I'm looking for "tail-bound-like" inequalities that look like this (I state a specific setting but more general settings are interesting):
Let $D$ be a distribution on a set of "nice" functions $g$:...
9
votes
1
answer
886
views
Concentration of sum of powers of normals
Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables i.e.~$X\sim\...
1
vote
0
answers
274
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Concentration bound for $f(w) = w \times \sin wz$
I need to find an exponential bound for $P(|S_n - \mu| > \lambda)$ where $S_n = \frac{1}{D} \sum_{i=1}^D w_i \sin w_iz$ for a constant $z$, $E(S_n) = \mu$ and $w_i$ are drawn from the normal ...
-1
votes
2
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605
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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 ...
4
votes
1
answer
349
views
Variance of maximum of mixture of gaussians
Let $\{X_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X_1,\ldots,X_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some symmetric,...
5
votes
2
answers
684
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Asymptotic Expansion of Distribution in Central Limit Theorem for Non-Identically Distributed Random Variables
My question is related to the following theorem (e.g. Section XVI.4 of Feller's 1971 book): Let $Z_i$ $(i=1,\cdots,n)$ be independent and identically distributed random variables with mean zero, ...
1
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0
answers
98
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Small ball probabilities for functions of correlated normals
Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
2
votes
1
answer
274
views
Is there monotonicity of measure concentration?
Suppose $X$ and $Y$ are nonnegative random variables such that $\mathrm{Pr}(X\geq t)\leq\mathrm{Pr}(Y\geq t)$ for all $t\geq0$. Now take $X_1,\ldots,X_n$ to be independent with the same distribution ...
1
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0
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611
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Upper bound on expectations of the sum of product of a martingale difference sequence with a predictable sequence, weighted by certain random weights
Let $(\mathcal{F}_i)_{i\geq 1}$ be a filtration. Let $0\leq p_i\leq 1$, be a random variable measurable w.r.t. $\mathcal{F}_i$. Consider two sequences of random vectors $v_i\in\mathbb{R}^M,w_i\in\...
2
votes
2
answers
907
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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
$u_k\ge 0$
$\|u_k\|_{L^1}=\int u_k=1$...
2
votes
1
answer
439
views
Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure
Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define
$$\Delta(u)= \frac{\int u(h) \exp(-\eta u(h))\exp(-\frac{\lambda}{2}h^2)~\mathrm{d}h}{\...
5
votes
1
answer
765
views
Measure concentration for weakly dependent random variables
For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a zero-...
2
votes
2
answers
291
views
How many boxes so that there is $k$ of same of color from $n$ different colors?
Say you have $m$ boxes each of which is colored with one of $n$ colors. What should $m$ be so that the probability that there is atleast $k$ boxes with one same color is strictly greater than $\frac{1}...
6
votes
2
answers
720
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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 $(\epsilon_{ij})_{ij}$ ...
0
votes
2
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492
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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 $u$ ...
6
votes
0
answers
337
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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:
http://terrytao.wordpress.com/2010/01/03/254a-...
3
votes
1
answer
1k
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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 $\epsilon > 0$. ...
7
votes
2
answers
2k
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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] = e^{-\Theta(t^...
4
votes
1
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503
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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 ...
10
votes
1
answer
931
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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 ...
5
votes
3
answers
903
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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 \mathbb{R}^...
1
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0
answers
205
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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
1
answer
647
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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 $\mathcal{...
5
votes
0
answers
397
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 $p(U,V)$...
11
votes
2
answers
3k
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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 \...