Questions tagged [measure-concentration]
The measure-concentration tag has no usage guidance.
397 questions
2
votes
0
answers
140
views
Matrices with i.i.d. Heavy tail Columns
I'm wondering if there are any known results about minimum eigenvalue of matrices with i.i.d. heavy tailed columns. In particular, Theorem 5.62 of Roman Vershynin's notes (http://www-personal.umich....
- 859
7
votes
2
answers
2k
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] = e^{-\Theta(t^...
- 1,791
1
vote
0
answers
34
views
Limiting law of quadratic functions of sample averages
Let $X_1,\cdots,X_n$ be independent centered univariate random variables. Let also $\{w_{ij}\}_{i,j=1}^{k,n}$ be a set of deterministic scalar weights, where $k\ll n$. Define sample averages
$$
\...
- 373
3
votes
0
answers
193
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-...
- 31
6
votes
2
answers
720
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}$ ...
- 293
1
vote
0
answers
361
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$ ...
- 39
3
votes
1
answer
460
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 ...
3
votes
0
answers
268
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 ...
user76479
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-...
- 23k
1
vote
0
answers
124
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 ...
0
votes
0
answers
102
views
Probability of random variable being lesser than the other
Say there are two independent random variables, $X$ and $Y$, and we have samples $\{x_1,\dots x_n\},\{y_1,\dots y_n\}$. I am interested in bounding the probability of the event $C = \mathbb{1}_{X<Y}...
- 197
5
votes
3
answers
903
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 \mathbb{R}^...
- 1,010
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 ...
- 13.9k
1
vote
0
answers
146
views
minimum eigenvalue of Katri-Rao product of two Gaussian matrices
Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum ...
- 363
3
votes
1
answer
339
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 ...
- 13.9k
2
votes
0
answers
246
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 (...
- 599
5
votes
0
answers
169
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$...
- 461
0
votes
2
answers
492
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$ ...
- 320
2
votes
2
answers
907
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 u_k=1$...
- 320
5
votes
0
answers
327
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 ...
- 161
5
votes
1
answer
284
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 $\mathcal{N}(0,\mathbf{I}/2)+i\mathcal{N}(0,\...
- 859
3
votes
1
answer
1k
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$. ...
- 298
4
votes
0
answers
76
views
How well does an estimator perform on another dataset?
Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$.
Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\...
- 515
0
votes
1
answer
196
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 ...
- 1,352
3
votes
1
answer
247
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, ...
- 205
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}...
- 225
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,...
- 941
2
votes
0
answers
124
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 ...
- 461
-1
votes
2
answers
605
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 ...
- 461
-1
votes
1
answer
104
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,$ $|...
- 277
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}...
- 13.9k
5
votes
0
answers
138
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 ...
- 121
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}{\...
- 323
1
vote
0
answers
611
views
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\...
- 323
2
votes
0
answers
145
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 ...
- 57
1
vote
0
answers
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 $x\in A$ ...
- 5,474
1
vote
0
answers
274
views
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 ...
4
votes
1
answer
503
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 ...
- 461
2
votes
0
answers
64
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-...
- 121
5
votes
0
answers
137
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}[...
- 619
3
votes
0
answers
152
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^{...
- 859
1
vote
0
answers
112
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 ...
- 111
6
votes
0
answers
337
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:
http://terrytao.wordpress.com/2010/01/03/254a-...
- 7,633
5
votes
0
answers
117
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}...
- 859
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 $p(U,V)$...
- 2,449
1
vote
0
answers
98
views
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: ...
- 111
1
vote
0
answers
205
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 ...
- 965