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Questions tagged [asymptotics]

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0answers
22 views

Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions (i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$ (ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
7
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2answers
792 views

Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that $$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$ ...
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0answers
69 views

Rate of convergence for difference between conditional and marginal probability

Suppose $X\sim \text{Bin}(2n,p)$ and $X_1,X_2\sim\text{Bin}(n,p)$ are independent, with $X_1+X_2=X$. I'm interested in the rate of convergence for the absolute difference $$ \left\vert P(X>c|X_1\...
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2answers
108 views

Are cyclic codes bounded by a continuous function?

In coding theory, we know that if you take the function \begin{equation} \alpha_q(\delta) := \limsup_{n \rightarrow \infty} \ \max \{ R(C) \mid \delta(C) \ge \delta \mid C \subseteq \mathbb{F}_q^...
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1answer
79 views

Is there any solution that currently exists for the graph automorphism problem in the general case?

I was reading the Wikipedia pages on the graph automorphism, but I could not find any solution to the problem (Not even a brute force one). So, is it indeed true that no solutions exist for the ...
6
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1answer
140 views

How did they come up with the MRRW bound?

Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is Suppose $C \...
7
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0answers
134 views

Has this self-similar sequence the ratio $(\sqrt2+1)^2$?

This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows: $a_n$ is the smallest number such that $s_n:=...
2
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0answers
132 views

a Kernel free asymptotic for a sampling operator

Let $\Pi=\left\{ t_{k}\right\} _{k\in\mathbb{Z}}$ a sequence of real numbers such that $-\infty<t_{k}<t_{k+1}<+\infty$ for every $k\in\mathbb{Z}$, $\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty$...
3
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0answers
88 views

Friedlander-Iwaniec Flipping moduli

I am reading section 12 (Flipping Moduli) of the paper "The polynomial $X^2+Y^4$ captures its primes" by Friedlander and Iwaniec. At page 997, just below equation (12.7) we start estimating the ...
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0answers
32 views

Steepest descent integration in several dimensions

The method of steepest descent provides an asymptotic approximation for integrals of the form: $$I = \int_C \exp(M f(z))\mathrm dz$$ for large positive $M$, where $f(z)$ is analytic in the region of ...
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1answer
59 views

Asymptotic upper bound for partial binomial-like sum

I want to upper bound the quantity $$\sum_{i\le \alpha n} \binom{n}{i}\lambda^i$$, where ${\lambda>1}$, $0<\alpha<1$. It is not the same as partial sum of binomial coefficients. An asymptotic ...
3
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1answer
82 views

Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$

Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
1
vote
1answer
140 views

Deriving condition to get correct asymptotic bound

Suppose that $X\sim \text{Bin}(n,\theta)$. Note that $X$ is the sum of $n$ $iid$ Bernoulli($\theta$) random variables. By the local limit theorem (Theorem 7 here) for the sum of discrete random ...
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0answers
67 views

Generalization of regularly varying functions

A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$, $$ \lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a) $$ for some function $g(a)&...
2
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1answer
95 views

Asymptotics for the first zero of the Bessel functions

Let $J_\nu$ be the standard Bessel function of the first kind and let $x_\nu$ be its smallest zero. Is there a simple reference or result for the asymptotic expansion of $x_\nu$ when $\nu$ goes to $+\...
6
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1answer
179 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 ...
3
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1answer
215 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
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1answer
105 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
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1answer
336 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
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1answer
192 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
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0answers
67 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
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1answer
105 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
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1answer
120 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
240 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)^{...
1
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1answer
141 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
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0answers
26 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
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1answer
268 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
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0answers
83 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
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0answers
23 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
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1answer
209 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
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1answer
123 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 ...
5
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0answers
151 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
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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
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1answer
43 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: $\...
3
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0answers
196 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
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1answer
117 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
1answer
70 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
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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}(...
4
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0answers
38 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
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1answer
82 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
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1answer
254 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
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1answer
239 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
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1answer
316 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
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0answers
84 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
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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
144 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
114 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
126 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
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0answers
25 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
112 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)) ~ \...