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1
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0answers
20 views

Concentration of functional of Gaussian random variable

Suppose I have two Gaussian distributions $p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...
2
votes
1answer
52 views

How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question. Given a ...
6
votes
2answers
94 views

Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables

Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...
1
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0answers
56 views

asymptotic behavior of Lipschitz constants of sectional curvature

I'm studying the paper "Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds" (https://arxiv.org/pdf/1402.4947v1.pdf) and I have some problem in understanding the proof of ...
3
votes
1answer
77 views

Tail bound for product of normal distribution

Let $U, V$ be two standard normal random variables with covariance $cov(U,V) = \beta \in [0,1)$. Let $W = UV$ be the product of two RV's, and $W_1, W_2, \ldots, W_n$ be n i.i.d copies of $W$, what's ...
2
votes
1answer
73 views

Symmetry of concentration bounds on mean

Question summary: If I have a two-sided bound, can I immediately get a one-sided bound with tighter constants? Question details: Let $\mathbf X = X_1,...,X_n$ be $n$ i.i.d. real-valued random ...
5
votes
1answer
194 views

Concentration bounds on weighted sum of i.i.d. Bernoulli random variables

Let $X_1,\dots, X_n\sim\operatorname{Bern}(\frac{1}{2})$ be independent, identically distributed random variables, and $\alpha=(\alpha_1,\dots,\alpha_n)\in[0,1]^n$ a vector of non-negative weights ...
2
votes
0answers
127 views

concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
7
votes
1answer
277 views

Level sets of weakly differentiable funtions

Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose $$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$ where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
0
votes
0answers
50 views

Matrix concentration inequality

Let $X \in \mathbb{R}^{n \times d}$ be a fixed matrix and $W \in \mathbb{R}^{n \times d}$ be a random matrix with elements $w_{ij} = x_{ij} + \epsilon_{ij}$, where $\epsilon_{ij}$ are iid subgaussian ...
4
votes
1answer
56 views

Concentration of U-statistics for exchangable distributions (and the unbounded case)

Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If $|h(w_1,...
4
votes
2answers
169 views

Expected value of Bernoulli quadratic forms

Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
1
vote
1answer
83 views

Upper tail concentration of sample covariance matrices

I'm interested in concentration of the following random matrix sum in spectral norm $\frac{1}{m}\sum_{k=1}^m b_k^2\mathbf{a}_k\mathbf{a}_k^*$ Here $\mathbf{a}_k\in\mathbb{R}^n$ are i.i.d. standard ...
1
vote
0answers
67 views

Concentration of the quotient of random variables

Let $X_1, X_2, \cdots, X_n$ be n i.i.d. standard Gaussian random variables. It is clear that we can describe the concentration of $\sum_{i=1}^n \alpha_i X_i$, and $\sum_{i=1}^n \alpha_i X_i^2$ (sub-...
2
votes
0answers
34 views

tail bounds for sum of n iid variables divided by power of n

Let $X_i, 1\leq i\leq n$ be i.i.d. random variables with finite moments. Then $Y_n :=\frac{1}{n^{1+\delta}}\sum_{i=1}^nX_i$ goes to 0 almost surely for any $\delta >0$. What are some good non-...
7
votes
2answers
198 views

concentration inequality for entropy from sample

Consider a measure $\mu$ on a finite set, and let $x_1, \ldots, x_n$ be i.i.d samples from $\mu$. Then the expression $S_n = -\frac{1}{n} \sum_{i=1}^n \log \mu(x_i)$ converges by a.s. to the entropy $...
0
votes
0answers
41 views

Existence of optimal coupling in optimal transport

Let $P,Q$ be any two distributions over a space $\mathcal{X}$ and let $\mathcal{M}(P,Q)$ be the set of all couplings of $P$ and $Q.$ For a given metric $d$ over $\mathcal{X},$ the optimal transport ...
9
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2answers
353 views

Adaptive version of the Azuma–Hoeffding inequality

The Azuma inequality states that if we have a martingale $X_1,\ldots,X_N$ that satisfies a bounded difference condition: $$|X_k - X_{k-1}| \leq c_k$$ Then: $$\Pr\left[X_N - X_0 \geq \sqrt{2\sum_kc_k^2 ...
4
votes
1answer
207 views

Tail bounds on eigenvalue gaps for GUE

What I'm looking for is a non-asymptotic bound on the probability that the smallest gap between eigenvalues of a GUE matrix does not exceed a certain value. I'm aware of the bounds in http://imrn....
2
votes
1answer
166 views

Asymptotic behavior of a ratio of sums of iid random variables

Let $X_i$ and $Y_i$ be distributed identically to $X$ and $Y$, respectively. Assume both $X$ and $Y$ take strictly positive values. Consider the random variable $R_n \doteq \frac{\sum_{i=1}^n X_i}{\...
2
votes
0answers
99 views

Intuitive (?) inequality extremal inequality

Consider $N$ pairs of random variables $(X_i, Y_i)$. $X_i$ are iid, with $EX_i=0$ and $EX_i^2=1$. The same conditions hold for $Y_i$. Moreover all $X_i$ are independent of all $Y_j$. It seems very ...
8
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1answer
802 views

Doob Martingale: Where is the catch?

I am working on a research problem in uncertainty propagation that involves sums of possibly dependent random variables with bounded sets of support. I am attempting to use the method of bounded ...
3
votes
0answers
112 views

McDiarmid-like inequality for subgassian random variables

Let $X_n$ be a set of $N$ subgaussian random variables, not necessarily independent, with $E\exp(\lambda X_n) \le \exp(\lambda^2/2)$. Let $X=(X_1,\ldots, X_N)$ and $f:\mathbb R^N \rightarrow \mathbb R$...
2
votes
0answers
125 views

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(...
3
votes
0answers
131 views

A generalization of coupon collector problem - $\geq1$ pick per experiment

Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back. $N=1$ is standard coupon ...
4
votes
1answer
97 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
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0answers
102 views

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 $Sign(n_i) = -1$ ...
4
votes
2answers
118 views

Do subgaussian variables obey the slightly-stronger-than-Chernoff tail bound?

If $X \sim Normal(0,1)$, then we have the tail bound: $$ (*) \qquad\Pr[X > t] \leq \mathcal{O}\left(\frac{e^{-t^2/2}}{t}\right) .$$ Now for general variables $X$, a nice condition is that $X$ be ...
2
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0answers
133 views

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 (...
2
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1answer
139 views

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
0answers
81 views

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 ...
2
votes
0answers
230 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 ...
7
votes
4answers
187 views

What can be said about the concentration of measure of product of Gaussian variables?

I have a set of random variables $X_1,\ldots,X_n$, all Gaussian with mean 0 and variance 1, indepedent. Let $p(x_1,\ldots,x_n)$ be some polynomial that takes products and sums of $x_1,\ldots,x_n$. ...
3
votes
1answer
112 views

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, ...
0
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0answers
117 views

What is the concentration of measure for Gaussian random variables which are independent, but are transformed?

This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for). The question is split into two: I have a matrix $X ...
0
votes
1answer
93 views

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 $|\nabla u_j|\rightharpoonup d\mu,$ $|...
2
votes
0answers
98 views

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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0answers
111 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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0answers
109 views

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
1answer
116 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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0answers
64 views

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
1answer
96 views

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 ...
1
vote
1answer
555 views

Hoeffding's inequality for vector valued random variables

Is there a version of Hoeffding's inequality for vector valued random variables? This seems to be hard to find and I wonder why. I suppose it is difficult to show Hoeffding's lemma, since the proof ...
4
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0answers
104 views

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
1answer
412 views

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)...
2
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0answers
98 views

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^{...
3
votes
1answer
159 views

Strictly positive solutions of a random linear system

Suppose $B\in\mathbb{R}^{m\times n}$ is a random binary matrix with i.i.d entries and $c\in \mathbb{R}^m$ is a strictly positive vector, that is $c_i>0$ for $i=1,2,\cdots m$. Also assume $m<n$, ...
4
votes
0answers
93 views

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}...
3
votes
2answers
357 views

Measure concentration for law of large numbers

The classical law of large numbers states that $$\frac1k\sum_{i=1}^k X_i \rightarrow \mathbb{E} X_1$$ for i.i.d. $X_1, X_2, \ldots$ with finite $L^1$ norm. I was wondering whether is it possible to ...
8
votes
1answer
271 views

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 $\...