Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [concentration-of-measure]

Use this tag for questions about the principle that a random variable that depends in a Lipschitz way on many independent variables (but not too much on any of them) is essentially constant.

Concentration of measure (about a median) is a principle applied in measure theory, probability and combinatorics and has consequences in other areas such as Banach space theory. Informally, the principle is that a random variable that depends in a Lipschitz way on many independent variables (but not too much on any of them) is essentially constant.

Following are links for learning more.

431 questions
12
votes
2 answers

Lipschitz function of independent sub-Gaussian random variables

If $X\sim \mathcal{N}(0,I)$ is a Gaussian random vector, then Lipschitz functions of $X$ are sub-Gaussian with variance parameter 1 by the Tsirelson-Ibragimov-Sudakov inequality (eg. Theorem 8 here). Suppose $X = (X_1,X_2,\ldots, X_n)$ consisted of…
Hedonist
  • 1,401
1
vote
0 answers

Let $X$ be a random variable with a finite mean $\mu$ and $E[|X−\mu|^n] < ∞$. Find $a$ s.t $P(X ≥ \mu + c) ≤ \frac{E[|X − \mu|^n]}{a}$ ($c > 0,n > 0$)

I am trying to apply Markov inequality here. $P(X ≥ \mu + c) = P(X - \mu ≥ c) ≤ \frac{E[X − \mu]}{c}$, but I cannot figure out where does $E[|X − \mu|^n$ come from. Since there is an absolute value, I am also thinking about Chebyshev’s…
FUFU
  • 11
0
votes
0 answers

Concentration on a unit sphere

I am trying to understand measure concentration on a unit sphere, using this text book. Lectures on Discrete Geometry by Jiří Matoušek https://link.springer.com/book/10.1007/978-1-4613-0039-7 On page 331, there is a theorem (Measure concentration…
user1168149
  • 9
0
votes
0 answers

Question about X concentrates to its mean

Suppose for a statistic $X$ of a size-n sample, $E|X-EX|\le f(n)$ for some decreasing function $f(n)$. Can we say $X$ concentrates to its mean at a rate of $f(n)$? I understand concentration rate is often defined by how the tail probability…
Justin
  • 67
0
votes
0 answers

Can the Bernstein concentration inequality bound for matrices be improved when we have a sequence of diagonal matrices?

I am using the Bernstein inequality for the sum of independent random Hermitian matrices presented in the following paper by Joel A.Tropp (page 96, theorem 6.6.1) which states the following: Consider a finite sequence $\{X_k\}$ of independent,…
Imelda Lacroix
  • 1
0
votes
1 answer

Continuous function $f \colon S^n \to [0,1]$ that is not concentrated

I would like to construct a continuous function from a sphere to the unit interval $f\colon S^n \to [0,1]$ that is not concentrated. In other words, for every $n$ and every $x\in[0,1]$, the set $f^{-1}(x-\frac{1}{3},x+\frac{1}{3})$ has measure at…
pizet
  • 1,163
  • 1
  • 9
  • 19