All Questions
Tagged with pr.probability measure-concentration
324 questions
10
votes
4
answers
645
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 ...
- 3,946
7
votes
2
answers
594
views
Large deviation/concentration inequality for submartingale
Let $S_t = M_t + D_t$ be the sum of a martingale $\left(M_t\right)_{t=1,2,\ldots}$ and a predictable process $(D_t)_{t=1,2,\ldots}$ such that the variance of the increments of $M$ is uniformly bounded ...
- 3,968
18
votes
1
answer
1k
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 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 is ...
- 8,071
4
votes
0
answers
141
views
Is there an example that both Berry-Essen bound and DKW bound are attained?
The Berry-Essen bound stated that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$
where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
- 8,071
2
votes
1
answer
508
views
Extension of Gordon's comparison inequality to subgaussian processes?
"Theorem A" in this paper by Y. Gordon:
http://www.math.uiuc.edu/~mjunge/Comps/Gordonm.pdf
is a comparison inequality for Gaussian processes:
Is there an analogue of this result for subgaussian ...
- 517
3
votes
0
answers
125
views
Concentration of sums of random matrices around the mean, in the Loewner order
Recently, I have found myself interested in concentration properties of random matrices.
Specifically I would like to answer questions of the following sort
Let $\{X_i\}_{i=1}^n$ be i.i.d. copies ...
- 393
9
votes
0
answers
1k
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 ...
- 618
4
votes
1
answer
347
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^\...
CommunityBot
- 1
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
0
answers
759
views
Product of two random Gaussian matrices - orthant probability
Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...
- 968
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
6
votes
0
answers
554
views
a variation on Hanson-Wright inequality
The classic Hanson-Wright inequality states that for a Gaussian random vector $\mathbf{x}\in\mathbb{R}^n$ distributed as $\mathcal{N}(\mathbf{0},\mathbf{I})$ and $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...
- 859
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
0
answers
451
views
concentration bounds on weighted multinomial sum
Consider i.i.d random vectors $Y_{1},..,Y_{n}$ and they are chosen uniformly at random from $\{e_{1},..,e_{L}\}$ where $e_{i}$ is a $L\times 1$ vector with $i$th component be 1 and the others be 0. ...
- 31
4
votes
1
answer
286
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 ...
CommunityBot
- 1
9
votes
1
answer
1k
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 ...
- 47
3
votes
1
answer
282
views
Longest runs and concentration of measure
Consider the longest runs $\ell_\sigma(x)$ of the pattern $\sigma$ for $\sigma\in \{0, 1, 01, 10, 001,\dots\}$ etc. in a binary sequence $x=x_1\dots x_n$.
For example, $\ell_{001}(0001110010011001)=2$...
- 23.2k
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
10
votes
2
answers
847
views
Minimum separation among $m$ random points on an $n$-dimensional unit sphere
Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...
CommunityBot
- 1
5
votes
1
answer
331
views
Matrix concentration bound
Suppose we have $N$ constant matrices $A_i \in R^{m\times m}, 1\leq i \leq N$. Consider $N$ random rotation-matrices $R_i \in SO(m), 1\leq i \leq N$. Is it possible to obtain a concentration bound on
$...
- 166
1
vote
0
answers
295
views
One-sided Talagrand concentration inequality for empirical processes
Let $\mathcal{F}$ denote a function class. A classic result by Talagrand states that
\begin{align*}
\mathbb{P}\bigg\{\sup_{f\in\mathcal{F}}\big|\sum_{i=1}^nf(X_i)-\mathbb{E}\big[\sum_{i=1}^nf(X_i)\...
- 859
12
votes
0
answers
489
views
Is this extension of Hoeffding's inequality known?
Question Overview:
Is it already known that, when using Hoeffding's inequality to lower bound the mean of i.i.d. random variables, you can replace the upper bound on the random variables with the ...
- 399
4
votes
1
answer
502
views
Hoeffding's inequality for sums of pairs of random variables
Let $X_1,\dotsc, X_n$ be $n$ i.i.d. random variables where $X_1 \in [a,b]$. Similarly, let $Y_1,\dotsc,Y_m$ be $m$ i.i.d. random variables where $Y_1 \in [c,d]$. Furthermore, $X_i$ and $Y_j$ are ...
- 128k
2
votes
1
answer
271
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 ...
7
votes
2
answers
606
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/...
- 24.2k
2
votes
1
answer
172
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 ...
- 399
4
votes
1
answer
681
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 ...
- 128k
6
votes
1
answer
3k
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 ...
CommunityBot
- 1
4
votes
0
answers
416
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{...
- 859
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
8
votes
2
answers
486
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 $...
- 128k
5
votes
2
answers
575
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 \...
CommunityBot
- 1
9
votes
2
answers
1k
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 ...
- 28k
3
votes
2
answers
589
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 ...
4
votes
1
answer
355
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....
- 114k
2
votes
1
answer
886
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}{\...
- 1,687
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
3
votes
1
answer
1k
views
concentration inequality for averages of dependent random variables
Let $X \in R^n$ be a random vector such that
$$P(|X_i| > \epsilon) > e^{-\epsilon^2}$$
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| ...
- 461
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
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 ...
CommunityBot
- 1
2
votes
0
answers
366
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(...
CommunityBot
- 1
1
vote
0
answers
360
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
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
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 ...
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
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
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
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 ...
CommunityBot
- 1
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
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