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Yesterday the following question was asked by user sigmatau: I'm interested in the following question: given $n$ i.i.d. random variables $X_i \sim \mathcal{N}(0,\sigma^2_1), i=1,\ldots,n$ …

Let $b_1,b_2,\dots$ be positive real numbers such that $$s_1<\infty\quad\text{and}\quad z_1<\infty, $$ where $$s_k:=\sum_{j=k}^\infty b_j\quad\text{and}\quad z_k:=\sum_{j=k}^\infty\frac{b_j}{\sqrt{ …

The following question was asked at https://mathoverflow.net/questions/360053/asymptotic-growth-bound-on-a-sequence-equivalent-to-an-asymptotic-growth-bound-o, but then deleted by the user: I post …

Can one find the asymptotics of $f_d(a,b)$ for $a\in(0,1)$ and $d\to\infty$, with $p$ fixed and (say) uniformly in $a\in(0,1)$ and $b\in(b_{\min},b_{\max})$, where $b_{\min}:=a^{2-2/p}(1\wedge d^{2/p- … As pointed out in answers here and here, an appropriate asymptotics of $f_d(a,b)$ would be enough to determine the asymptotics of the surface area of a unit $\ell_p^d$-ball for $d\to\infty$ -- an apparently …

$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$. Let us say that a family $(r_j)_{j\in J}$ of r …

Turán's theorem says the following. Take any natural $n$ and $r$. Suppose that \begin{equation*} |G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0} \end{equation*} where $|G|$ is the number of edges of an ( …

What is $$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N} |2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$ where $\mathbb N:=\{1,2,\dots\}$? In other words, I would like to know …

$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let \begin{equation*} \exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^ …

$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then \begin{equation} \om(G)\ge …

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant $$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i) $$ over all $a_0,\dots,a_{n-1}$ such that $0 … =a_0<\dots<a_{n-1}=1$ (or, better, an upper bound on $M_n$ which is asymptotic to $M_n$; or, at least, the asymptotics of $\ln M_n$). …

What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$, $$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$ and $$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\sqr …