# Tag Info

## Hot answers tagged measure-concentration

13 votes
Accepted

### Violating the Lebesgue density theorem

It is a theorem of Besicovitch that measures on $\mathbb R^d$ do satisfy the density theorem. Fremlin, Measure Theory, Chap. 47 added Besicovitch, around 1930, extended his density ...
• 38.8k
13 votes
Accepted

• 55.5k
6 votes
Accepted

### Tail probability of random projection

$\newcommand{\R}{\mathbb{R}} \renewcommand{\P}{\operatorname{\mathsf P}} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta}$ In view of the spherical symmetry of the distribution of the $l$-...
• 90.3k
6 votes
Accepted

### Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Without loss of generality, $R=1$. Let $Z_1,\ldots,Z_n$ be iid standard normal random variables (r.v.'s). Then \sqrt n\, X_1\overset{\text{D}}=\frac{\sqrt n\,Z_1}{\sqrt{Z_1^2+\...
• 90.3k
6 votes

### Concentration inequality for the law of iterated logarithm

As was noted in the comments by Yuval and Kevin, even if $X_1$ is bounded, the best upper bound on the probability in question is a negative power of $\ln n$. To get such a bound (and even an ...
• 90.3k
6 votes
Accepted

• 90.3k
6 votes
Accepted

• 90.3k
5 votes

• 90.3k
5 votes
Accepted

### Concentration of sum of concentrated random variables

There is a bad news and a good news. The bad one is that if you have no information other than that the probability of the $\varepsilon$-deviation is at most $p$ for each variable, then you can hardly ...
• 55.5k

Only top scored, non community-wiki answers of a minimum length are eligible