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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 ...
9
votes
4answers
483 views

More than $n$ approximately orthonormal vectors in $R^n$

This question was asked at math.stackexchange, where it got several upvotes but no answers. It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$. However, it is well established ...
8
votes
3answers
763 views

Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$

Hi. Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I mean by this ...
8
votes
2answers
348 views

Supremum of measure of sets of measure less or equal to 1/2.

Let $(X,d)$ be a metric space equipped with a probability measure $\mu$ (defined on the Borel $\sigma$-algebra on the topology induced by the metric $d$). I am interested in the different values that ...
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 ...
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 ...
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 ...
7
votes
1answer
215 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 ...
7
votes
1answer
175 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 ...
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 ...
6
votes
1answer
413 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 ...
6
votes
4answers
101 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 $x_1,\ldots,x_n$. ...
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 $(\epsilon_{ij})_{ij}$ ...
5
votes
1answer
182 views

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 ...
5
votes
0answers
81 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 ...
5
votes
0answers
242 views

Balls and bins — concentration bounds pertaining to the minimal load bin

Consider the standard balls and bins process, where $m$ balls are thrown uniformly at random into $n$ bins. Previous work has been done on estimating the value of the maximum load (i.e., the number of ...
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
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
2answers
567 views

Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)

Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is $$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$ ...
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 ...
4
votes
1answer
261 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 ...
4
votes
2answers
92 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 ...
4
votes
1answer
82 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 ...
4
votes
1answer
190 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 $i \ne j$ ($p$ ...
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 ...
4
votes
0answers
80 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
0answers
99 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 ...
3
votes
1answer
176 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 ...
3
votes
1answer
143 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 $m<n$, ...
3
votes
1answer
137 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 ...
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 $\epsilon > 0$. ...
3
votes
0answers
62 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 ...
3
votes
1answer
89 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, ...
3
votes
0answers
85 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*} ...
3
votes
2answers
363 views

Non-asymptotic large deviations for a convex set

Let $X_1,\dots,X_n$ be $n$ i.i.d random variables taking values in a Polish vector space $\mathcal{X}$ and with (Borel) probability distribution $\mu$. For any convex, compact $\Gamma \subset ...
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 ...
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| ...
2
votes
1answer
136 views

What is the spectral norm of a random projection times a diagonal?

Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix. What can be said about the distribution of the largest ...
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 ...
2
votes
1answer
310 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 ...
2
votes
1answer
320 views

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
1answer
267 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 ...
2
votes
1answer
308 views

A Johnson-Lindenstrauss lemma for finite fields?

Given $m$ points in $\mathbb{R}^N$, the Johnson-Lindenstrauss lemma guarantees the existence of a linear operator $\mathbb{R}^N\rightarrow\mathbb{R}^n$ that nearly preserves pairwise distances between ...
2
votes
0answers
74 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 ...
2
votes
0answers
95 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
votes
1answer
102 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 ...
2
votes
0answers
215 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 ...
2
votes
0answers
74 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 ...
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 ...
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 ...