# Tag Info

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
Accepted

• 55.5k
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
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 \begin{equation} \sqrt n\, X_1\overset{\text{D}}=\frac{\sqrt n\,Z_1}{\sqrt{Z_1^2+\...
• 90.3k

### 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
Accepted

• 90.3k
Accepted

• 90.3k

The claimed inequality is not true. The simplest possible counterexample works: let $x,y \in \mathbb{R}^n$ with $|x-y| = \epsilon$, and take $\mu = \delta_x$, $\nu = \delta_y$. Then $W_1(\mu,\nu) = \... • 28.1k 5 votes Accepted ### Good upper-bound for$\mathbb E[|X-np|^r]$where$X \sim \text{Binomial}(n,p)$and$r \ge 1$By the main result of the paper Exact Rosenthal-type bounds, we have $$E|X-np|^r\le c^r E|\Pi_\lambda-\lambda|^r$$ for real$r\in(2,\infty)\setminus(3,4)\setminus(4,5)$, where real$c>0$and$\...
• 90.3k
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 ...