The asymptotics tag has no usage guidance.

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143 views

### Asymptotic Expansion of Bessel Function Integral

I have an integral:
$$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$
and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically ...

**4**

votes

**0**answers

93 views

### Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...

**4**

votes

**1**answer

92 views

### Bounds for the size of arrays with distinct subarray sums

Consider an array $A$ of length $n$ with $A_i \in \{1,\dots,s\}$ for some $s\geq 1$. For example take $s = 6$, $n = 5$ and $A = (2, 5, 6, 3, 1)$. Let us define $g(A)$ as the collection of sums of all ...

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votes

**1**answer

307 views

### Interesting behaviour of binomial coefficients

Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...

**4**

votes

**1**answer

178 views

### On approximation of $\sum_{a,b=1}^n\gcd(a,b)$

Denote $g(n)=\sum_{a,b=1}^n\gcd(a,b)$, can we prove that
$$g(n)=\frac6{\pi^2}n^2\ln n+Cn^2+O(n\ln n)$$, where $C=-\frac12+\frac{6}{\pi^2}(-\frac12+\gamma-\ln(2\pi)+12\ln A),$ where $\gamma$ denotes ...

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votes

**0**answers

62 views

### Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$

Setup
This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms.
So, let $Z$ be a $p$-dimensional random vector with (unknown) ...

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votes

**1**answer

100 views

### Cardinality of growth rates

$Let f=o(g)$ stand for $\lim_{x\to\infty} f(x)/g(x)=0$.
It is simple to prove the following fact.
Proposition. Let $f_0,f_1:\mathbb{R}\to(0,\infty)$ be such that $f_0=o(f_1)$. Then there is a family ...

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votes

**0**answers

52 views

### Asymptotics for sum involving Euler numbers

This first request may be easy, but the asymptotics for the next step has me scratching my head.
Through an informal inductive argument I have been able to show
$$ (1) \quad \sum_{j=0}^{n-1}2^{2m(n-j)...

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votes

**1**answer

216 views

### Asymptotic behavior of a certain trigonometric partial sum

Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum:
$$
f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...

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votes

**0**answers

25 views

### Rate of convergence of centered Hotelling's statistic to Chi-squared distribution

Consider the Hotelling's statistic $H_n := n\mu_n\Sigma_n^{-1}\mu_n$, where $\mu_n$ (resp. $\Sigma_n$) is the empirical mean (resp. empirical covariance matrix) of a zero-mean random $d$-dimensional ...

**1**

vote

**1**answer

132 views

### Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$

Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R_n := \mathbb E_{\hat{P}_n}[Z]^T\...

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vote

**0**answers

25 views

### Positive density of certain arithmetic sequence

Let $\{a_i\}_{i=0}^{\infty}$ be a sequence of positive real numbers bounded above by some uniform constant. For each $k,n\geq 1$ let
$$
A^{(k)}_n:=\prod_{i=k}^{n+k-1}a_i
$$
and suppose that there ...

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votes

**1**answer

262 views

### Asymptotic behavior of sum linked with Lagrange interpolation

I already asked this a few weeks ago with no answer, so let me formulate differently.
In performing Lagrange interpolation with nodes 1/n, one encounters the sum
$$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{...

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votes

**0**answers

78 views

### Ratio of solutions to two heat equations

Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...

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**0**answers

18 views

### Convergence of the asymptotic expansion solution of homogeneous linear ODE of order 2

Consider ODE $w''+pw'+qw=0$, $p$ and $q$ are functions of $z$. Denote $w_1$ and $w_2$ the ODE's two linear independent solutions. As $z\to0$, in which situation:
Do both $w_1$ and $w_2$'s ...

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votes

**1**answer

202 views

### Finding an asymptotic solution for a first order ODE

Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...

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vote

**1**answer

117 views

### Upper bound of the fraction of gamma functions

Is there a simple upper bound of the following fraction of gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...

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votes

**0**answers

22 views

### Asymptotic Constancy of solutions of delay/integro differential equations

I have found quite a few papers on asymptotic constancy of solutions of delay differential equations and integral differential equations (see e.g. this reference or this reference).
I am however most ...

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votes

**0**answers

146 views

### $X$-rays of permutations

Consider the set of permutations $\mathfrak{S}_n$, on $\{1,2,\dots,n\}$, and identify each element $\pi\in\mathfrak{S}_n$ with the corresponding permutation matrix.
There has been some study (e.g. ...

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votes

**1**answer

33 views

### Right tail decay of F distribution [closed]

Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$?
$$\mathbb{P}(X\geq x)$$
what is the order of the above probability as $x\to+\infty$?

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vote

**1**answer

30 views

### Asymptotic eigenvalue distribution of sum of two i.i.d random matrices with Marchenko Pastur distributed eigenvalues?

Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where:
$\...

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votes

**0**answers

185 views

### What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?

What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...

**3**

votes

**1**answer

113 views

### Tail probability of random projection

Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...

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votes

**1**answer

68 views

### How many hyper-rectangle-like objects are intersecting a hyperplane?

Let $A\in \mathbb R^{n\times n},\ b\in \mathbb R^n$ such that $\forall x\in \{-1,1\}^n : Ax\ne b$.
Let us denote: $S=\{x\in\mathbb R^n|Ax=b\}$ ('S' for solution set).
Is $\ \#\Big\{H\in\big\{ \{-1\},...

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votes

**2**answers

64 views

### Left tail of convex combinations of $\chi_1^2$

Suppose $a_1,...,a_n\geq0, \sum_{i=1}^na_i=1$ and $Z_1,...,Z_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order?
$$\mathbb{P}(...

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votes

**0**answers

31 views

### Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...

**1**

vote

**1**answer

80 views

### What is the order of the left tail of a mixture of non-central chi-square?

Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean.
Let $X\sim\exp(\lambda)$ where the ...

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votes

**1**answer

247 views

### Asymptotic solution for a first order ODE

Simplified question*:
Given $f(t)$ that satisfies $f'(t)>0$, $f'(t)=\omega\left(t^{-1}\right)$, $\log\left(f'(t)\right)=o\left(f(t)\right)$ we denote $F=\exp\left(f\left(t\right)\right)$. Let $H(...

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votes

**1**answer

234 views

### Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambda(m)=\...

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votes

**1**answer

311 views

### Approximation a sum involving log and binomial coefficient

I am wondering about the asymptotic approximation of the following expression:
$$S=\sum^{N}_{i=0}\log\Bigg[\binom{\binom{N+1}{i}}{t_i}\Bigg]$$
where
$$t_i=\binom{N}{i}-\binom{N-k}{i-k}+\binom{N-k}{i-...

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votes

**0**answers

81 views

### Concerning some Tauberian-type asymptotics of Laplace transform involving $e^{-\sqrt{s}}$

There are some well-known Tauberian theorems concerning the asymptotics of the original function (say as $t$ tends to $0$) and that of its Laplace transform (as $s$ tends to infinity). I want to ask a ...

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votes

**0**answers

29 views

### How many points are still contained in a common (hyper-)ellipsoid?

It is known that
$${d+2\choose 2}-1$$
points uniquely determine a quadric in $\Bbb R^d$. However, I want my points not on an arbitrary quadric, but on a centered hyperellipsoid in $\Bbb R^d$, or ...

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votes

**1**answer

142 views

### How to estimate a summation?

For $v, w \in \{0,1\}^n$, denote $v w = (v_1 w_1, \ldots, v_n w_n)$ and $|v|=\sum_{i} v_i$.
Let $v_1, v_2 \in \{0,1\}^n$ and
\begin{align*}
f(x_1, x_2) = \sum_{d=0}^{|v_1 v_2|} \frac{1}{2^{|v_1|+|...

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votes

**0**answers

113 views

### How to compute the asymptotic of a summation which involves binomial coefficients?

Let $v_1,v_2 \in \{0,1\}^n$. Denote $v_1v_2=((v_1)_1 (v_2)_1, \ldots, (v_1)_n (v_2)_n)$ and $|v|=\sum v_{i}$.
\begin{align}
{\scriptsize
f(v_1, v_2) = \sum_{x_1=0}^{|v_1|} \sum_{x_2=0}^{|v_2|} \sum_{d=...

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votes

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125 views

### Gradient blowup for the 1-dim heat equation near an irregular boundary point

I'm trying to estimate the rate of boundary gradient blow-up for the 1-dim heat equation near an irregular boundary point.
Let $b(t) := (1-t)^\alpha$, $\alpha < 1/2$. Let $u(t,x)$ solve the ...

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**0**answers

24 views

### critical points relevant to the lowest order non-perturbative correction

I am interested in the Hyperasymptotics of multidimensional integrals of the form
$$\mathcal{I}(\lambda) = \int_{\mathbb{R}^n} dz_1 \wedge dz_2 \wedge \dotsi \wedge dz_n \, g(z_1,\dotsi,z_n) \, e^{\...

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votes

**3**answers

104 views

### Asymptotic forms of Legendre functions for large degree

Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are
$$
P_n(\cosh(x)) ~
\...

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votes

**2**answers

173 views

### Expectation of minimum of correlated Gaussian

What is the order of the following expectation with respect to $n$?:
$$\mathbb{E}(\min_{1\leq i\leq n}|z_i|^2)$$
where
$$(z_1,...,z_n)^T\sim N(0,I+11^T), 1=(1,1,...,1)^T$$
I know that when $z_i$ are ...

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vote

**1**answer

124 views

### $\lim_{k\to\infty}{n\choose k}2^{1-{k\choose2}}$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$

How do you prove $\lim_{k\to\infty,k\in\mathbb{N}}{n\choose k}2^{1-{k\choose2}}=1$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$? The expression ${n\choose k}2^{1-{k\choose2}}$ ...

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votes

**1**answer

150 views

### The Gauss Circle Problem asymptotic in dimension

The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...

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votes

**1**answer

292 views

### Asymptotic behavior of an integral depending on an integer

A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where
$$
f(n) := \...

**5**

votes

**2**answers

191 views

### Asymptotic rate for $\sum\binom{n}k^{-1}$

This MO question prompted me to ask:
What is the second order asymptotic growth/decay rate for the sum
$$\sum_{k=0}^n\frac1{\binom{n}k}$$
as $n\rightarrow\infty$?

**3**

votes

**1**answer

135 views

### Distribution of eigenvalues of a Wishart matrix

Is there a known expression for the eigenvalue distribution of a matrix of the form
$$\sum\limits_{i=1}^n k_ia_ia_i^T$$
where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)...

**0**

votes

**1**answer

88 views

### Compare $\operatorname{rad}(an+b)$ and $\varphi(cn+d)$ in a simple and interesting inequality, for some choice of integers $a,b,c$ and $d$

We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...

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votes

**3**answers

345 views

### Expected cardinality of a randomly chosen element of the family of subsets of $\{1,\ldots,n\}$ with at most $k$-elements

Assume that $1\le k \le n$ and let $\mathscr{Z}$ be the family of all subsets of $\{1,\ldots,n\}$ with at most $k$ elements. Pick a random element $X$ of $\mathscr{Z}$ (we consider the probablity ...

**1**

vote

**1**answer

132 views

### Asymptotic behavior of a solution of an ODE

I am not quite sure if this question is appropriate for this site as it might be not of a research level.
I am interested in the following ordinary differential equation on the real line
$$f’’(x)+(x-...

**0**

votes

**1**answer

142 views

### When does this recurrence stop?

Denote $n_i=n_{i-1}-\sqrt[k]{n_{i-1}}$. If $n_0=n$ then what is the minimum $i$ at which $n_i<2$ holds? Is there a standard technique to solve such problems? Any references?

**6**

votes

**4**answers

261 views

### Limit of a Combinatorial Function

I need help with the following problem, proposed by Iurie Boreico:
Two players, $A$ and $B$, play the following game: $A$ divides an $n \times n $ square into strips of unit width (and various ...

**0**

votes

**0**answers

25 views

### Are there any non-asymptotic bounds for the minimum empirical risk vs theoretical risk?

I'm trying to see if there's any bounds on the difference between $f_{ERM}$ and $f^{*}$. For now, define $\mathcal{F}$ to be a function class.
Let $P$ be a probability measure and $\hat{P_n}$ be the ...

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vote

**0**answers

101 views

### On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Let
$$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$
where $\zeta$ denotes the Riemann zeta function.
What are the reasonable asymptotic estimates for $I(T)...