All Questions
Tagged with pr.probability measure-concentration
324 questions
2
votes
1
answer
847
views
Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix
Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (not necessarily idempotent)
what is the distribution of ${y^T M y}$?
is there a high probability bound on $|{y^T M y}|$?
Most ...
- 128k
3
votes
1
answer
196
views
Uniform Convergence for Vectors
$\textbf{Problem statement:}$
Let $\mathcal H:\mathcal X \rightarrow \{0,1\}$ be a class of Boolean functions for $\mathcal X \subset \mathbb R^n$, and let the VC Dimension of $\mathcal H$ be $VC_{...
- 63.9k
12
votes
2
answers
2k
views
Can we do better than Azuma-Hoeffding when the variance is small?
The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
- 10.4k
8
votes
2
answers
1k
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 bound:...
- 128k
1
vote
1
answer
137
views
Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$
Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the ...
- 128k
4
votes
1
answer
206
views
Inner product of sorted Gaussian vector
Suppose $X_1,\ldots,X_n$ are i.i.d. standard normal. I'm wondering how to analyze the following quantity:
$$\left|\frac{X_{(1)}X_{(n)}+X_{(2)}X_{(n-1)}+\cdots+X_{(n)}X_{(1)}}{n}\right|$$
where $X_{(1)}...
CommunityBot
- 1
1
vote
2
answers
327
views
Use covering number to get uniform concentration from pointwise concentration
Let $\Theta$ be a subset of a metric space. Suppose $(X_\theta)_{\theta \in \Theta}$ is a random process on $\Theta$ which is $L$-Lipschitz and with the property that there exists constants $A, B>0$...
- 6,853
9
votes
2
answers
4k
views
What is (approximately) the expected value of $X\log{ X}$ where $X$ is binomial (or Poisson)?
Let $X$ follow a binomial distribution with parameters $n$ and $p$. Are there any known bounds for the expected value of $X\log{X}$, for large $n$ and small (but fixed) $p$? A Poisson approximation ...
- 128k
6
votes
1
answer
446
views
Is there a counterexample to the Thin Shell Conjecture for sub-exponential distributions?
The thin shell conjecture states that there exist universal constants $C,c>0$ such that every logconcave isotropic random vector $X$ in every Euclidean space $\mathbb{R}^n$ satisfies
$$\mathbb{P}\...
- 7,633
0
votes
1
answer
239
views
Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$
Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$.
Question
Given $\epsilon > 0$ (may be assumed to be very small), what is ...
- 6,853
1
vote
1
answer
467
views
Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$
Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$.
Question
What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(...
- 128k
1
vote
0
answers
686
views
Almost orthogonality of independent random vectors [closed]
If $X_1$ and $X_2$ are two independent isotropic random vectors in $\mathbb{R}^n$,
then $\mathbb{E}\|X_i\|_{2}^{2}=n$, $\mathbb{E}\langle X_1,X_2\rangle^{2}=n$.
How can I show from the above result ...
- 137
2
votes
1
answer
180
views
Generalization of Komlós–Major–Tusnády Approximation
The Komlós–Major–Tusnády Approximation (see Wikipedia) considers the sum of uniform variables in $(0,1)$. There are also version where instead the sum of equiprobable $0/1$ variables is used ($p=1/2$)....
- 128k
11
votes
1
answer
1k
views
Maximal inequality for the average of i.i.d. random variables
Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and $\mathbb{E}|Z_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like ...
- 61.9k
1
vote
1
answer
118
views
What is the order of the left tail of a mixture of non-central chi-square?
Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean.
Let $X\sim\exp(\lambda)$ where the ...
CommunityBot
- 1
4
votes
0
answers
1k
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 ...
- 3,968
5
votes
0
answers
1k
views
Asymptotic behavior of row sums in 2-d array of random variables
Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables:
$B^m_{1,1}$ $B^...
- 61
2
votes
1
answer
326
views
What is the Wiener measure of the curves with Hölder index $\frac 1 2$?
One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...
- 5,407
10
votes
2
answers
455
views
Largest deviations for uniform order statistics
Let $n >0$.
Let $X_1,\ldots,X_n$ be i.i.d. uniform random variable on $[0,1].$ Denote by $X^{(1)}\leq X^{(2)} \leq \cdots \leq X^{(n)}$ their order statistics, and write $\Delta^{(i)} = \vert X^{(...
3
votes
1
answer
178
views
Tail probability of random projection
Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...
- 128k
0
votes
1
answer
254
views
Does log-concave approximable distribution satisfy transportation-cost inequality?
Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition
$$
\...
- 11.4k
7
votes
1
answer
424
views
Transportation-cost inequality for pushforward measure
Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...
- 128k
3
votes
0
answers
166
views
Matrix Chernoff sampling with out replacement
I am interested to know if the matrix Chernoff bound (see Theorem 5.1.1 in https://arxiv.org/pdf/1501.01571.pdf) holds if one samples without replacement.
For example, the Bernstein inequality is ...
- 34.7k
1
vote
0
answers
175
views
Matrix Bernstein for spherical random variables
Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
- 111
4
votes
0
answers
162
views
Are sums extremal for subgaussian concentration?
Bobkov and Houdre https://projecteuclid.org/euclid.bj/1178291721
showed that among all $f:R^n\to R$ that are $1$-Lipschitz
with respect to the $\ell_1$ metric,
the variance is maximized by sums. ...
- 6,541
5
votes
1
answer
299
views
Variance modulo 1
The fact that the variance of the sum of independent random variables is the sum of their variances allows one to have a good understanding of how well-concentrated each term $X_i$ in a sum of $n$ ...
- 61.9k
3
votes
1
answer
294
views
Concentration inequalities specialized for log-likelihood / log-density functions
Let $P$ be a probability measure and $f$ be some probability density function (not necessarily related to $P$). Consider the function
$$
L(X_1,\ldots,X_n)
=\frac1n\sum_{i=1}^n\log f(X_i),
\quad
X_i\...
- 6,541
1
vote
0
answers
34
views
Limiting law of quadratic functions of sample averages
Let $X_1,\cdots,X_n$ be independent centered univariate random variables. Let also $\{w_{ij}\}_{i,j=1}^{k,n}$ be a set of deterministic scalar weights, where $k\ll n$. Define sample averages
$$
\...
- 373
4
votes
0
answers
93
views
On symmetry and measure concentration rate for convex bodies
The concentration of measure on the cube $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$,
states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $,
we ...
- 365
1
vote
0
answers
676
views
Are Outer Products of Sub-Gaussian Vectors Sub-Exponential?
$\newcommand\xx{\mathbf{x}}\newcommand\yy{\mathbf{y}}\newcommand\A{\mathbf{A}}\newcommand\aalpha{\boldsymbol{\alpha}}\newcommand\bbeta{\boldsymbol{\beta}}\newcommand\E{\mathbb{E}}\newcommand\inner[1]{\...
- 153
4
votes
1
answer
290
views
On the 1/2 assumption on concentration of measure for continuous cube
The concentration of measure on $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$,
states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $,
we have:
$$...
- 128k
4
votes
1
answer
1k
views
Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences
A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:
Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
- 128k
2
votes
1
answer
167
views
Concentration of emperical conditional probability
Assume sequence $(X_1,X_2, X_3, \ldots)$ is a first-order Markov sequence of real random variables where $X_i \in \mathcal{X}$ for some alphabet $\mathcal{X}$ of finite size $k$. Define emperical ...
- 482
7
votes
1
answer
466
views
Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?
It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds:
$$
ℙ\left(\left|
\sum_{i=1}^N a_i X_i
\right| \ge t \right)
\le
2\...
- 4,713
5
votes
0
answers
543
views
Vector martingale concentration
Let $\varepsilon_1, \dots, \varepsilon_N$ be a martingale difference sequence in $R^d$ with $\|\varepsilon_n\| \le B_n, a.s.$ for each $n=1,\dots,N$. Do we have some Azuma-type concentration ...
- 719
3
votes
2
answers
731
views
Concentration inequality for sum of iid random variables that involve KL distance
Conider $X \in \mathbb{R}^d$ and $Y \in \{0,1\}$, and a joint distribution $p_{XY}(x,y)$, and a set of $N$ i.i.d. samples $\{(X_i,Y_i)\}_{i=1}^{N}$. Define $p_{X0} = p_{XY}(x,0)$ and $p_{X1} = p_{XY}(...
- 128k
4
votes
1
answer
229
views
Product of estimates of mean values - Concentration of measure inequality
Let $X_{1},...,X_{d} \in \{-1,1\}^d$ be random variables, with $E[X_j]=\mu_j$. Having $n$ i.i.d. samples $x^{(i)}_1,x^{(i)}_2,....,x^{(i)}_d$, $i=1,...,n $, let $\hat{\mu}_{j}=\frac{1}{n}\sum^{n}_{i=1}...
- 7,249
1
vote
0
answers
376
views
Anti-concentration bounds for folded normal and inverse of gaussian variables
Are there any easy to use bounds on sums of the following kind :
$$
\sum_{i = 1}^{i = N} |a_i| \geq P \\
a_i \sim \mathcal{N}(0, 1) \\
$$
and also for sums of the form :
$$
\sum_{i = 1}^{i = M} \...
- 111
1
vote
1
answer
249
views
On concentration of a sum random variable
Take a random variable defined as
$$r=u_{11}v_{1}v_{1}+u_{12}v_{1}v_{2}+\dots+u_{n,n-1}v_{n}v_{n-1}+u_{nn}v_{n}v_{n}$$ where $v_{i}$ are independent uniform random variables from $\{0,\dots,b\}$, $u_{...
- 128k
2
votes
0
answers
323
views
McDiarmid's Inequality bounding deviation with multiplicative error?
Fix $m$ arbitrary values $x_1, x_2, ..., x_m$ in $[0,1]$, and an integer $n$. Obtain $n$-set $S$ by drawing $n \le m$ times randomly without replacement from $\{1,2,..,m\}$. Define r.v. $X = \sum_{i ...
CommunityBot
- 1
3
votes
0
answers
77
views
A concentration problem of product of matrices
Let $A$ be an $n \times m$ matrix with non-negative entries and $B \in \mathbb{R^{n\times n}_{\geq 0}}, C \in \mathbb{R^{m\times m}_{\geq 0}}$ be random matrices where B and C are both symmetric and ...
- 195
4
votes
0
answers
162
views
Concentration Inequality for Score Functions of Exponential Familty
Let $p$ be the density of a continuous one-parameter exponential family distribution on $\mathbb{R}$. We assume that
$$p(x) = c(x)\cdot \exp\bigl [ x \cdot \theta - b(\theta ) \bigr ], $$
where $\...
- 1,127
2
votes
0
answers
60
views
Mean width of intersection of two elipsoid
My question is regarding mean widths. For a set $\mathcal{T}$ define the mean width
\begin{align*}
\omega(T)=\mathbb{E}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I})}\bigg[\underset{\mathbf{u}\in\mathcal{...
- 363
5
votes
0
answers
143
views
Is there a concentration inequality depending on dimension for a symmetric function on product space?
I recently read an elegant paper of Bobkov
Bobkov, S.G., On concentration of measure on the cube, J. Math. Sci., New York 165, No. 1, 60-70 (2010); translation from Probl. Mat. Anal. 44, 55-64 (2010)....
- 593
1
vote
0
answers
110
views
Tail bound without independence
Suppose $X_i , X_j\in \mathbb{R}^d$ are gaussian vectors and $A$ is an $n\times n$ symmetric PSD matrix where $A_{ij} = f(\|X_i-X_j\|_2), \quad i,j\in 1,\ldots,n\;$ for some non-negative Lipschitz ...
- 195
1
vote
1
answer
122
views
Variance bound of a functional
$X_1,\ldots,X_n$ are i.i.d standard normal random variables.
$a_1,\ldots, a_n$ are constants with $a_i \in [\kappa_1, \kappa_2]$ for all $i$ and $\kappa_1>0$.
$\hat c_n$ is given as the solution ...
5
votes
1
answer
372
views
What are some of results in low dimensional statistics that do not hold in high dimensions?
This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? and current heated research front of high dimensional ...
- 8,071
3
votes
0
answers
186
views
Anti-concentration for sum of t-wise independent uniform variables
Let $X_{1},\ldots,X_{n}$ be i.i.d. random variables, each variable is uniform over the set of integers $\{ 0,\ldots,D-1 \}$. Let $S = \sum_{i=1}^{n}X_{i}$.
By ``small ball probability'', we have that ...
- 225
5
votes
1
answer
295
views
Constructive Central Limit Theorem
Background: Let $\{a_i\}_{i=1}^n$ be i.i.d. random variables with zero-mean and unit variance, on a probability space $\Omega$. Define $$s_n=\frac{1}{\sqrt{n}}\sum_{i\leq n} a_i$$
Central limit ...
- 24.2k
2
votes
2
answers
1k
views
Lower bound on number of samples for an epsilon delta approximation matching the Chernoff bound
So we have two biased coins, one comes out head w.p. $1/2+\epsilon$ and the other w.p. $1/2-\epsilon$. How many times should we flip these two coins to be able to tell them apart w.p. at least $\delta$...
- 6,541