All Questions
Tagged with measure-concentration pr.probability
23 questions
5
votes
3
answers
5k
views
Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere
Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
- 6,853
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
14
votes
3
answers
2k
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 ...
- 539
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 $\|...
- 1,955
10
votes
3
answers
4k
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
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$
...
- 223
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
$$
\rho = \min_{i,j\in{...
- 1,127
8
votes
2
answers
1k
views
Does Multiplicative Version of Azuma's Inequality Hold?
It is known that there are multiplicative version concentration inequalities for
sums of independent random variables. For example, the following
multiplicative version Chernoff bound.
Chernoff bound:...
- 175
7
votes
3
answers
496
views
Chernoff-type bounds for a stopped sum of independent random variables
Let $Y_1, \ldots, Y_n$ and $X_1, \ldots, X_n$ be i.i.d. $p$-Bernoulli random variables and let $T \in \{0, \ldots, n\}$ be a stopping time for the process. From Wald's equation, we know
$$
E\left[\...
- 153
6
votes
3
answers
447
views
Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace
Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
- 6,853
5
votes
1
answer
1k
views
Explicit constant for Carbery–Wright inequality
The Carbery–Wright inequality is a seminal result about the anti-concentration of polynomials of Gaussian random variables.
See e.g. Meka, Nguyen, and Vu - Anti-concentration for polynomials of ...
- 111
4
votes
1
answer
474
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 $g$:...
- 4,529
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
4
votes
0
answers
143
views
For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?
Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the ...
- 1,955
3
votes
3
answers
5k
views
Hoeffding's inequality for vector valued random variables
Is there a version of Hoeffding's inequality for vector valued random variables?
This seems to be hard to find and I wonder why. I suppose it is difficult to show Hoeffding's lemma, since the proof ...
- 619
3
votes
1
answer
182
views
How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?
Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...
- 6,853
2
votes
1
answer
212
views
Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$
Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...
- 6,853
2
votes
1
answer
936
views
Upper-bound for spectral norm of the covariance matrix of a certain Gaussian vector with correlated entries
Let $n$ and $m$ be large positive integers. Let $x=(x_1,\ldots,x_n)$ be a vector of independent random variables from $N(0,1)$. It is clear that the covariance matrix of $x$ is $I_n$, the identity ...
- 6,853
2
votes
2
answers
690
views
Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample
Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$.
...
- 6,853
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
1
vote
1
answer
230
views
VC dimension of a certain derived class of binary functions
Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$...
- 6,853
1
vote
1
answer
216
views
Rademacher complexity of function class $(x,y) \mapsto 1[|yf(x)-\alpha| \ge \beta]$ in terms of $\alpha$, $\beta$, and Rademacher complexity of $F$
Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$...
- 6,853
1
vote
1
answer
415
views
Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
- 6,853
0
votes
1
answer
273
views
Sum of sequences of random variables, with variable success probabilities
Consider two sequences of (not necessarily independent) Bernoulli random variables $X_1, X_2, \ldots, X_n$ and $Y_1, Y_2, \ldots, Y_n$. Suppose that for any $i$, we have $\Pr[X_i = 1] = \Pr[Y_i = 1] = ...
- 153