All Questions
91 questions
6
votes
1
answer
203
views
Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$
Disclaimer. Question moved from SE.
Setup
Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$.
Question
What is a good upper-bound for $\mathbb E[|X-np|^r]$ ?
Solution for small $r$
If $r=2$, then ...
- 6,853
5
votes
0
answers
711
views
Concentration inequality for max component of a multivariate Gaussian in the general case
I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
- 283
1
vote
2
answers
250
views
Finite-sample deviation bound of empirical distribution from true distribution
Let $P=(p_1,\ldots,p_k) \in \Delta_k$ be distribution supported on set of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ based on an iid sample of size $n$.
Question
What's a good non-...
- 6,853
4
votes
1
answer
239
views
Uniform inequality of the form $\text{Proba}(\sup_{v \in [-M,M]^k}|p^Tv-\hat{p}_n^Tv| \le \epsilon_n) \ge 1 - \delta$
Let $M > 0$, $k$ be a positive integer, and $\mathcal V:=[-M,M]^k$. Finally, let $p \in \Delta_k$, (where $\Delta_k$ is the $(k-1)$-dimensional probability simplex) and let $\hat{p}_n$ be an ...
- 6,853
1
vote
0
answers
123
views
Sanov-type finite-sample bound on $KL(P\|\hat{P}_n)$
Let $P$ be a distribution on an alphabet of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ via $n$ i.i.d samples $a_1,\ldots,a_n \sim P$, i.e $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$.
...
- 6,853
1
vote
1
answer
313
views
Bounds on difference between "logsumexp" and variance?
Let $Z$ be a random variable with finite moment-generating function $M_Z(\theta):=E[e^{\frac{1}{\theta}Z}]<\infty$ for all $\theta > 0$, and for $\delta \in (0,1]$, define
$C_Z^\delta := \inf_{\...
- 6,853
3
votes
1
answer
826
views
concentration inequality for a weighted sum of independent but not identical binary variables
Let $\alpha\in[0,1]$ be a fixed constant, and let
$w,x\in[0,1]^n$ be two vectors such that $\sum_i w_i x_i=\alpha$.
Define $Y = \sum_i w_i X_i$, where $X_i \sim \operatorname{Bernoulli}(x_i)$, so it ...
- 617
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 ...
- 123
1
vote
1
answer
365
views
Lower-bound probability of non-centered quadratic form
Let $X\sim N(\mu,\sigma^2I)\in \mathbb{R}^n$ be a non-centered ($\mu\neq 0$) Gaussian vector with independent coordinates. I'm wondering if there is any sharp lower bound of the following probability:
...
- 1,495
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 ...
- 6,853
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)}...
- 1,495
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(...
- 6,853
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 ...
- 1,495
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^{(...
- 245
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
$$
L(X_1,\ldots,X_n)
=\frac1n\sum_{i=1}^n\log f(X_i),
\quad
X_i\...
- 710
4
votes
1
answer
1k
views
Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences
A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:
Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
- 53
7
votes
1
answer
466
views
Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?
It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds:
$$
ℙ\left(\left|
\sum_{i=1}^N a_i X_i
\right| \ge t \right)
\le
2\...
- 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}(...
- 482
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_{...
- 197
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}$$ where $v_{i}$ are independent uniform random variables from $\{0,\dots,b\}$, $u_{...
- 13.9k
1
vote
1
answer
122
views
Variance bound of a functional
$X_1,\ldots,X_n$ are i.i.d standard normal random variables.
$a_1,\ldots, a_n$ are constants with $a_i \in [\kappa_1, \kappa_2]$ for all $i$ and $\kappa_1>0$.
$\hat c_n$ is given as the solution ...
5
votes
1
answer
372
views
What are some of results in low dimensional statistics that do not hold in high dimensions?
This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? and current heated research front of high dimensional ...
- 8,071
3
votes
2
answers
319
views
Concentration inequality of joint event over time of a submartingale
Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as
$$
P(X_t-X_0 \leq -t) \leq exp\...
- 185
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
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 $\delta$...
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
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
$$
\rho = \min_{i,j\in{...
- 1,127
4
votes
1
answer
347
views
Concentration of functional of Gaussian random variable
Suppose I have two Gaussian distributions
$p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...
- 515
7
votes
2
answers
605
views
Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables
Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...
- 71
5
votes
2
answers
565
views
Concentration of U-statistics for exchangable distributions (and the unbounded case)
Consider the following so-called $U$-statistic of order 2: $$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,...
- 1,731
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 ...
- 363
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(...
- 121
2
votes
0
answers
386
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 \...
- 131
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
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 ...
- 1,324
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:...
- 175
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 ...
- 181