# Tag Info

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A set of points on the unit sphere in ${\Bbb R}^n$ with $\langle x,y\rangle \le \cos \theta$ for all distinct $x$ and $y$ is called a spherical code with minimum angle $\theta$. For $0<\theta < \pi/2$, Kabatiansky and Levenshtein gave an exponential upper bound (of the form $\exp(C(\theta)n)$) for the maximum number of points in such a spherical code. ...

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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 properties of sets to those of finite Hausdorff measure. source next: D. G. Larman, "A new theory of dimension", Proc. London. Math. Soc. 17 (1967) 178-192 ...

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$\newcommand{\ep}{\varepsilon}$ Let $X$ be any nonnegative random variable (r.v.) with finite mean $\mu>0$ and variance $\sigma^2<\infty$. For any real $u>0$, we have $\ln\frac xu\le\frac xu-1$ for all real $x>0$, whence $x\ln x\le\frac{x^2}u+x\ln\frac ue$ and \begin{equation*} EX\ln X\le\frac{EX^2}u+EX\ln\frac ue=\frac{\sigma^2+\mu^2}u+\mu\...

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Here's a step that seems nice enough to point out. It still leaves a parameter to pick, and I'm not sure it's ever better than applying Bernstein, but it does something different. We can get a probability bound in terms of how much $S_n$ exceeds the Renyi entropy $H_{\alpha}$ of $\mu$ (equivalently, worded in terms of the $\ell_{\alpha}$ norm of $\mu$), for ...

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An almost complete answer. First of all, indeed the $\lVert \alpha\rVert_2$-based bound mentioned in the question can be shown to be tight for many "simple" $\alpha$'s, such as balanced, or uniform/balanced on a subset of $m$ coordinates. However, it is not tight in general, and the right answer appears to be captured by a quantity, the $K$-functional ...

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The preprint "Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation" by Johann S. Brauchart, Edward B. Saff, Ian H. Sloan, Yu Guang Wang, and Robert S. Womersley gives the following result in Corollary 3.4: $\mathbb{E}[N^{2/d}\Theta_\text{min}]\to C_d = (\kappa_d/2)^{-1/d}\Gamma(1+\tfrac{1}{d})$ as $N\to\infty$ It also gives a bound on ...

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This looks like a weak law of large numbers, and in fact a strong law holds: I claim that $\liminf_{t \to \infty} \frac{S_t}{t} \ge \Delta$ almost surely, which implies the desired result. The key is to show that $\frac{M_t}{t} \to 0$ almost surely. Then we have $\frac{S_t}{t} = \frac{M_t}{t} + \frac{D_t}{t} \ge \frac{M_t}{t} + \Delta$ and can take the ...

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This is worked out in some detail in the paper of Fan, Grama and Liu, J. Math Anal. Appl. 448 (2017), 538-566 (see in particular Theorem 2.1 there, and the references). Unfortunately I do not have an open access to an electronic copy, and it doesn't seem to exist on arXiv.

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It appears you want to have the following: Let $X_1,\dots,X_n$ be independent zero-mean random variables (r.v.'s ) (or, more generally, martingale-differences) with $S_n:=X_1+\dots+X_n$, $B^2:=EX_1^2+\dots+EX_n^2$, and $M:=\frac1n\sum_1^n M_i$, where $M_i:=\text{ess sup}|X_i|$. Then \begin{equation*} P(S_n\ge x)\overset{\text{(?)}}\le\exp-\frac{x^2}{...

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