Questions tagged [measure-concentration]
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386
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Uniform concentration bound (function-valued random variable / continuous stochastic process)
I'm trying to consider a probability space $\Omega$ and
$f(x,\xi):\mathcal{X}\times\Omega\to\mathbb{R}$ (stochastic process over space? or function-valued random variable?), where $\mathcal{X}\subset\...
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RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)
Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
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Is it true that if a random vector has independent coordinates each bounded by $1$ then $P[ \|X\| \leq \epsilon\sqrt{n}] \leq (C\epsilon)^{n}$?
I'm studying Vershynin's well-written book on "High Dimensional Probability" and the third chapter on concentration of random vectors.
Exercise 3.1.7 from the book is the following.
Let $X =...
- 391
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A maximal inequality
Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. symmetric random variables, with $-1\leq X_i\leq 1$, $\mathbb{E}(X_i) =0$, $\mathbb{E}(X_i^2) = 1$. We have that:
$$
P\left(\bigcap_{k = 1}^{n}\frac{|\sum_{i = ...
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Concentration bound for a increasingly weighted sum of bernoulli random variables
Given $x_1,x_2,\ldots,x_n$ i.i.d. bernoulli random variables with $P(x_i=1)=\frac1n$. Given a constant $c=1+\frac{1}{m}, m\geq n$. Is there an explicit theorem that can derive a concentration argument ...
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Weak Borell-TIS inequality for a subgaussian process
It is a known fact (Borell-TIS inequality) that, given an almost surely bounded Gaussian centered process $X(t), t \in T$, where $T$ is a topological space, $$\mathbb{P}\{\sup_t X(t)-\mathbb{E} \sup_t ...
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Simultaneous Concentration of $\sum_{i = 1}^{n} X_i^2$ and $\sum_{i = 1}^{n} X_i$ with $X_i$ iid. Poisson
Consider $n$ independent Poisson(1)-distributed random variables $(X_i)_{1 \leq i \leq n}$.
This is a (hopefully more interesting) follow-up question to Super-exponential concentration for $\frac{\...
- 533
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Concentration inequalities for functions of random binary strings
Let $(X_1,\ldots,X_n)$ be a vector in $\{0,1\}^n$ drawn uniformly at random among all vectors with exactly $k$ $1'$s. I am interested in inequalities for tail probabilities for the random variables $X,...
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Concentration of measure on spheres with respect to a unitary of trace approximately zero
Cross-posted from MSE, where it hasn’t received any answer yet:
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
- 1,534
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Concentration result for self-normalized empirical process
In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
- 206
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Inner product of the spherical cap and Gaussian
Let $d\in \mathbb{N}$ and $\eta \sim N(0,I_d)$ where $N(0,I_d)$ is the gaussian distribution with the covariance matrix of $I_d$. Also, define a spherical cap as follows. Fix $v \in \mathbb{S}^{(d-1)}$...
- 139
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Concentration inequalities for leave-one-out sum
Let $X_1,...,X_n$ be iid random variables. Consider $f:\mathbb{R}\times\mathbb{R}^{n-1}\to\mathbb{R}$ such that $f$ is symmetric in the last $n-1$ variables. Our goal is to show that $\sum_{i=1}^n f(...
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How does Chernoff-Hoeffding bound with limited independence reduce to the usual generic CH bound with complete independence
As the title might suggest, I am referring to this paper https://www.cs.umd.edu/~srin/PDF/ch-bounds.pdf , titled : Chernoff-Hoeffding Bounds for Application with Limited Independence.
The theorem in ...
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Concentration bounds for sum of weighted sampling without replacement
Let $X$ be a collection of $2l$ non-negative numbers $X_1,X_2,\ldots,X_{2l}$. We draw $l$ weighted (proportional to values) samples without replacement from $X$. Let $S$ denote this set of $l$ samples....
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Tight Chernoff Concentration for Bernoulli(p) RV
I remember seeing a research paper on tight concentration of Bernoulli(p) random variable in terms of $p$.
What I mean is that they used a stronger upper bound for the MGF than $E[e^{s(X-p)
}]\leq e^{...
- 113