The asymptotics tag has no usage guidance.
6
votes
1answer
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
0answers
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
1answer
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 ...
6
votes
1answer
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
1answer
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 ...
0
votes
0answers
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) ...
3
votes
1answer
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 ...
4
votes
0answers
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
$$ (1) \quad \sum_{j=0}^{n-1}2^{2m(n-j)...
6
votes
1answer
216 views
Asymptotic behavior of a certain trigonometric partial sum
Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum:
$$
f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...
0
votes
0answers
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
1answer
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\...
1
vote
0answers
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 ...
14
votes
1answer
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
$$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{...
3
votes
0answers
78 views
Ratio of solutions to two heat equations
Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
0
votes
0answers
18 views
Convergence of the asymptotic expansion solution of homogeneous linear ODE of order 2
Consider ODE $w''+pw'+qw=0$, $p$ and $q$ are functions of $z$. Denote $w_1$ and $w_2$ the ODE's two linear independent solutions. As $z\to0$, in which situation:
Do both $w_1$ and $w_2$'s ...
5
votes
1answer
202 views
Finding an asymptotic solution for a first order ODE
Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...
1
vote
1answer
117 views
Upper bound of the fraction of gamma functions
Is there a simple upper bound of the following fraction of gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...
0
votes
0answers
22 views
Asymptotic Constancy of solutions of delay/integro differential equations
I have found quite a few papers on asymptotic constancy of solutions of delay differential equations and integral differential equations (see e.g. this reference or this reference).
I am however most ...
5
votes
0answers
146 views
$X$-rays of permutations
Consider the set of permutations $\mathfrak{S}_n$, on $\{1,2,\dots,n\}$, and identify each element $\pi\in\mathfrak{S}_n$ with the corresponding permutation matrix.
There has been some study (e.g. ...
0
votes
1answer
33 views
Right tail decay of F distribution [closed]
Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$?
$$\mathbb{P}(X\geq x)$$
what is the order of the above probability as $x\to+\infty$?
1
vote
1answer
30 views
Asymptotic eigenvalue distribution of sum of two i.i.d random matrices with Marchenko Pastur distributed eigenvalues?
Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where:
$\...
2
votes
0answers
185 views
What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?
What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...
3
votes
1answer
113 views
Tail probability of random projection
Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...
-1
votes
1answer
68 views
How many hyper-rectangle-like objects are intersecting a hyperplane?
Let $A\in \mathbb R^{n\times n},\ b\in \mathbb R^n$ such that $\forall x\in \{-1,1\}^n : Ax\ne b$.
Let us denote: $S=\{x\in\mathbb R^n|Ax=b\}$ ('S' for solution set).
Is $\ \#\Big\{H\in\big\{ \{-1\},...
3
votes
2answers
64 views
Left tail of convex combinations of $\chi_1^2$
Suppose $a_1,...,a_n\geq0, \sum_{i=1}^na_i=1$ and $Z_1,...,Z_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order?
$$\mathbb{P}(...
3
votes
0answers
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
1answer
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 ...
3
votes
1answer
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(...
3
votes
1answer
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,
$$\Lambda(m)=\...
4
votes
1answer
311 views
Approximation a sum involving log and binomial coefficient
I am wondering about the asymptotic approximation of the following expression:
$$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-...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
1answer
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|+|...
3
votes
0answers
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=...
3
votes
1answer
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 ...
0
votes
0answers
24 views
critical points relevant to the lowest order non-perturbative correction
I am interested in the Hyperasymptotics of multidimensional integrals of the form
$$\mathcal{I}(\lambda) = \int_{\mathbb{R}^n} dz_1 \wedge dz_2 \wedge \dotsi \wedge dz_n \, g(z_1,\dotsi,z_n) \, e^{\...
2
votes
3answers
104 views
Asymptotic forms of Legendre functions for large degree
Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are
$$
P_n(\cosh(x)) ~
\...
8
votes
2answers
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 ...
1
vote
1answer
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 ${n\choose k}2^{1-{k\choose2}}$ ...
3
votes
1answer
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))+\...
10
votes
1answer
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
$$
f(n) := \...
5
votes
2answers
191 views
Asymptotic rate for $\sum\binom{n}k^{-1}$
This MO question prompted me to ask:
What is the second order asymptotic growth/decay rate for the sum
$$\sum_{k=0}^n\frac1{\binom{n}k}$$
as $n\rightarrow\infty$?
3
votes
1answer
135 views
Distribution of eigenvalues of a Wishart matrix
Is there a known expression for the eigenvalue distribution of a matrix of the form
$$\sum\limits_{i=1}^n k_ia_ia_i^T$$
where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)...
0
votes
1answer
88 views
Compare $\operatorname{rad}(an+b)$ and $\varphi(cn+d)$ in a simple and interesting inequality, for some choice of integers $a,b,c$ and $d$
We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...
7
votes
3answers
345 views
Expected cardinality of a randomly chosen element of the family of subsets of $\{1,\ldots,n\}$ with at most $k$-elements
Assume that $1\le k \le n$ and let $\mathscr{Z}$ be the family of all subsets of $\{1,\ldots,n\}$ with at most $k$ elements. Pick a random element $X$ of $\mathscr{Z}$ (we consider the probablity ...
1
vote
1answer
132 views
Asymptotic behavior of a solution of an ODE
I am not quite sure if this question is appropriate for this site as it might be not of a research level.
I am interested in the following ordinary differential equation on the real line
$$f’’(x)+(x-...
0
votes
1answer
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
4answers
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
0answers
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 ...
1
vote
0answers
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)...
