# Questions tagged [asymptotics]

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### Existence of function $f$ such that $f(x) \sim \sum_{j \in \mathbb{N}} x^{1 - \frac{1}{j}}$

Is there a function $f$ on $\mathbb{R}$ such that as $x \to 0$, $$f(x) = \sum_{j=0}^N x^{1 - \frac{1}{j}} + o(x^{1- \frac{1}{N}}),$$ for every $N \in \mathbb{N}$? Heuristically there shouldn't be ...
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### Randomized version of Turán's theorem II

$\newcommand{\om}{\omega}$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 \om(G)\ge\...
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### Randomized version of Turán's theorem

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 ...
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### Approximating the partial sum of remainders function

This is a question related to the one I posted here, but I have found some more interesting and general results and thought here might be a better place to ask. Let $R_{k,N}$ denote the remainder of ...
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### Can the Bessel functions tend to a plane wave?

Can the Bessel functions tend to a plane wave? If I have this function: $$y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)$$ ...
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### Two-term recurrence relation

We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$ $$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n$$ ...
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### Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

I have a system of nonlinear Volterra integral equations of form $$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$ and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
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### Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function

Let's denote $F_{t_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t_u$ with $u$ degrees of freedom and $F_{t_v}(x)$ the cumulative distribution function of the t-distribution $t_v$...
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### Convergence result on Cornish Fisher expansion of binomial distribution

Since it is known that Cornish Fisher expansion of quantiles does not have guaranteed convergence for all distribution, I wonder specifically if any convergence result is known in literature for CF ...
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### For the following class of matrices, are the determinants invariant under permutations?

I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
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### lower bound for sum of the n factors of the inclusion exclusion principle

Suppose the following relation is established: $P\Bigl(A\cup B \cup C\cup D\Bigr) < P\Bigl(E\cup F\cup G\Bigr)$ based on boole's inequality, for each of the above probabilities we can have the ...
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### Tail bounds on random series in Hilbert space

Tail bounds on random series in Hilbert space Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$, $n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ ...
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### Asymptotics of ratios of polynomially recursive sequences

A sequence $a_n$ is said to be polynomially recursive (P-recursive) if it satisfies: $$p^{[r]}(n)a_{n+r}+\cdots+p^{[1]}(n)a_{n+1}+\cdots + p^{[0]}(n)a_n=0$$ where $p^{[i]}(t)\in \mathbb{Q}[t]$ are ...
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### Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$

To begin, let us set $$A_Q(n):=\sum_{d|n \\ d<Q}\mu(d)$$ If we fix $Q$ and let $n$ vary, we get a very surprising amount of cancellation. For instance, the trivial bound \begin{align*} \mathbb{E}_{...
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### Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says: (i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists an interval $I \subseteq \mathbb{R}$ containing $0$ and a smooth $f: I \to V$ such ...
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### Clarification for a statment from ArXiv:1812.07690 “Asymptotics of Nahm sums at roots of unity”

This is a cross-post of the same math.SE question to MO, thinking that is better suited here. My question is about Lemma 2.1 from the ArXiv:1812.07690 by D. Zagier and S. Garoufalidis which concerns q-...
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### The uniform “probability” on $\mathbf{N}$: What occurs beyond logarithmic density?

This is a follow-up to Question #47134. There is obviously no uniform probability distribution on $\mathbf{N}$ (or $\mathbf{Z}$); however, using the notion of amenability, you can show that any ...
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### How to obtain the asymptotics of Legendre polynomials directly from their generating function

I'm reading about Legendre polynomials for additional information since it is interesting to know! Moreover it would help me with a task I am working on. See https://math.stackexchange.com/questions/...
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### Asymptotic bound for $\sum_{x=0}^\infty \sum_{y=0}^\infty (x+y)^m e^{-\frac{x^2}{2i} - \frac{y^2}{2j}}$ for $i$ and $j$ large

Note: This question relates to two previous questions on math.stackexchange (1 and 2), neither of which had satisfactory answers after posting bounties. Whilst trying to count certain types of ...
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Suppose that I have a sequence of $n$ numbers $x_1, \ldots, x_n$ all taken from the real interval $[0,1]$. I am interested in minimizing the infinity norm of the vector $$v = \left( \frac{x_{1}}{x_2},... 1answer 172 views ### An asymptotic expansion of a infinite sum I am interested in the asymptotic expansion in t(t>0) when t\to 0^+ of the following series$$ \sum_{k\ge 0}e^{-k^{2/n}t} $$for integer n>2 (n=1 follows from Poisson summation formula ... 1answer 130 views ### Existence of an asymptote for g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right) Working with the Slater's inequality (compagnion of Jensen's inequality) I find this statement : Let f(x) be a continuous,twice differentiable function ,convex or concave and non constant on (0,\... 1answer 77 views ### Saddle point approximation of terms in a sum (asked in MSE, but received no attention) Suppose I need to compute a sum,$$ \sum_{n=0}^N a_n,$$each term of which involves an integral,$$a_n=\int e^{Nf(x)+ng(x)}dx. I am interested in the large-$... 1answer 55 views ### Distribution of line segment intersections in random pointsets let$P$be a set of$n$points that are uniformly distributet inside the unit square ore unit circle, and$L=\lbrace\ell_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le ...
Let $E_1, \dots, E_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A_N = \frac{1}{N} ( 1_{E_1} + \dots + 1_{E_N} )$ be the proportion of the events $E_i$ that ...