Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable?

Let $X_0, X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of ...
cgmil's user avatar
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Another large noise limit

Note: Here all processes take values in $[0, 1]$. Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant. Let $X$ be the solution to the SDE $$dX_t = \sigma X_t \, dW_t$$...
Nate River's user avatar
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Closeness of a rational approximation

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 ...
Iosif Pinelis's user avatar
6 votes
2 answers
259 views

Asymptotics of error function integral with square root

I am interested in the asymptotics of the integral $$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$ for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
Julian's user avatar
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1 answer
140 views

Sum of an arithmetic sequence involving Euler factors

I am trying to find an asymptotic formula for the following sum as $T \to \infty$. $$ \sum_{t = 1}^{T} \prod_{\substack{p \; \textrm{prime} \\ p | t}} \rho(p) \frac{1 - \frac{1}{p^2}}{1 - \frac{\rho(p)...
Melanka's user avatar
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On an asymptotic integral

Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that $$ \int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}. $$ Does it follow that $\phi$ is a ...
Ali's user avatar
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17 votes
1 answer
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How hard is it to say "not exactly $p$" with a Horn sentence?

EDIT: immediately after bountying the question (whoops ...) I found, while looking for something else entirely, that Sauro Tulipani gave an explicit algorithm for producing a Horn sentence $\varphi_p$ ...
Noah Schweber's user avatar
1 vote
0 answers
148 views

Number-theoretic proof of integrality of a fraction and asymptotics of sum over partitions related to symmetric group

Consider $\;\alpha=(\alpha_1,...,\alpha_n)\in\mathbb{Z}_+^n\;$ such that $\;1\alpha_1+...+n\alpha_n=n.\;$ Let $\varphi$ denote Euler totient-function. Let $\;T_\alpha\;$ be a set of permutations in $...
te4's user avatar
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Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number

Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
Nilotpal Kanti Sinha's user avatar
5 votes
2 answers
179 views

Limit of the extremal process of i.i.d. Gaussians see from the tip

I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
MikeG's user avatar
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The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$ [closed]

An informal investigation of a sum. Consider this sum: $$S =\sum_{k=2}^{\infty}(k^{1/k} -1)$$ Does this converge? How does it behave as it diverges, if it diverges? If $k$ equaled $1$ we would get $0$ ...
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2 votes
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403 views

Weakly dependent central limit theorem

Say I have $N$ random variables $X_1,\cdots,X_i,\cdots,X_N$, with zero mean and finite variance. $X_i$ and $X_j$ are independent iif $|i-j|>m$, and positively correlated otherwise (say the ...
CWC's user avatar
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1 answer
259 views

Bounds on the number of integer compositions with parts bounded from above and below

I'm looking for asymptotic bounds (as n goes to infinity) on the number of integer compositions of $n$ with parts in $[a,n]$ and separately for parts in $[a,b]$, with $1 < a < b < n$. (To ...
blizzard's user avatar
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Branch cuts, inverse Fourier transform and large time asymptotics

Let the Fourier transform of $f(t)$ be defined as $F(\omega) = \int_{-\infty}^\infty dt f(t) e^{i\omega t}$ for values of $\omega$ where the integral exists. What are the precise conditions on $F(\...
Fetchinson0234's user avatar
2 votes
1 answer
92 views

Decay rate for a small perturbation of a simple linear ODE

MOTIVATION. Let $f:[0,+\infty)\to \mathbb{R}$. Solutions to $\partial_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero exactly as $e^{-\lambda t}$. This property is preserved if we apply an ...
Overflowian's user avatar
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Asymptotics through generating functions

I am working on the asymptotics of a sequence and wanted to use the method of subtracted singularities (Darboux's method) for its generating function $f$. But it turns out that the function has $1$ as ...
EED's user avatar
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2 votes
1 answer
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Identity involving double sum with binomials

(asked previously in MSE here) In the course of a calculation, I have met the following complicated identity. Let $A$ and $a$ be positive integers. Then I believe that $$ \sum_{B\ge A,b\ge a} (-1)^{a+...
Marcel's user avatar
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2 votes
0 answers
120 views

Average length of consecutive integers which have an increasing number of divisors

Consider the nine consecutive natural numbers starting from $1584614377$. ...
Nilotpal Kanti Sinha's user avatar
5 votes
2 answers
203 views

Integrals can sometimes be computed through their saddle points. Are there examples of converse, when saddle points are found via integrals?

Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g. $$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$...
Weather Report's user avatar
1 vote
1 answer
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Estimates of product of eigenvalues gaps for Wigner matrices

Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$. My question. For any fixed $i\in\{1,\dots,n\}$, ...
Ludwig's user avatar
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20 votes
1 answer
586 views

Distinct exponents in the factorization of the factorial, a problem of Erdős

In the 1982 paper below, Paul Erdős proved that if $h(n)$ is the number of distinct exponents in the prime factorization of $n!$ then $$c_1\Big(\frac{n}{\log n}\Big)^{1/2} < h(n) < c_2\Big(\frac{...
sebaztian's user avatar
  • 203
2 votes
1 answer
236 views

Maximum nearest neighbor distance for a Poisson point process

Is the maximum nearest neighbor distance between points of the process, over all the infinitely many points of a stationary Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, almost ...
Vincent Granville's user avatar
7 votes
2 answers
467 views

Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$

Let $f:\mathbb{R}\to\mathbb{R}$ be the function $$f(x) = \sum_{k=1}^{\infty} \sqrt{\max\left\{1 -\frac{k^2}{x^2},0\right\}}.$$ Numerical experiments suggests that there exists $n\in\mathbb{N}$ and a $...
Onur Oktay's user avatar
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5 votes
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200 views

Number of solutions of linear congruence with bounded variables

Fix some nonzero integers $a_1, \dots, a_k$, with $k \geq 2$, and some real numbers $c_1, \dots, c_k \in (0,1)$. For every positive integer $m$, let $N(m)$ be the number of $k$-tuple of integers $(x_1,...
Erik4's user avatar
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6 votes
1 answer
495 views

Rate of decrease of the Fourier transform of standard mollifiers

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|<\...
Iosif Pinelis's user avatar
2 votes
2 answers
226 views

Asymptotics of a delay differential equation

Say we have the line segment $L(t) = [0,t]$, and randomly remove open intervals of length $1$ from $L(t)$ until no more open intervals of length $1$ remain. Define $u(t)$ as the expected measure of ...
Christopher D. Long's user avatar
3 votes
0 answers
90 views

Probability of winning a $k$-rounds coin toss game

Let $p,q \in [0,1]$ with $p>q$. I denote by $B_k(p), B_k(q)$ two independent random variables following the binomial distribution, with parameters $(k,p)$ and $(k,q)$ respectively. Informal ...
Argemione's user avatar
  • 131
6 votes
1 answer
221 views

Convergence speed of the tail of distribution using Tauberian remainder theorem

This question may be related to this one. Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem. Let $f$ be some one-sided probability ...
Seung Hyeon Yu's user avatar
5 votes
1 answer
158 views

Asymptotics for repulsive aggregation(-diffusion) equation

Consider the aggregation-diffusion equation $$ \frac{\partial \rho}{\partial t} = \nabla (\rho \nabla(W\star \rho)) + \nu \Delta \rho, $$ where $W:\mathbb{R}^d \to \mathbb{R}$ is a twice continuously ...
Peter Koepernik's user avatar
4 votes
0 answers
176 views

As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?

As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
Likes Algorithms's user avatar
1 vote
1 answer
258 views

Asymptotic lower bound for the number of square free with at least two prime factors

In one of Soundararajan's papers, he claims without proof that it is a standard exercise to show that the number $N(X)$ of positive square-free integers $d \equiv 1 \; \bmod \; 8$ less than $X$, with ...
Melanka's user avatar
  • 577
1 vote
0 answers
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A sampling problem

I have a suspicion that the question I am about to ask is classical. I could not trace a reference, and I am really curious about the answer. Here is the question, An urn contains $m$ balls ...
Liviu Nicolaescu's user avatar
0 votes
0 answers
242 views

Singularity of inverse exponential integral function

The exponential integral function is defined by $$ Ei(z) = \int_{-\infty}^z dw \frac{e^w}{w} \,.$$ Away from the negative real axis the exponential integral function has a Taylor series about $z=0$: $$...
Samuel Crew's user avatar
3 votes
1 answer
103 views

Asymptotic growth of ternary partitions of integers $3n$

Consider the binary partitions of $2n$ in powers of $2$, denoted by $b(2n)$, with the generating function $$\sum_{n\geq0}b(2n)\,x^n=\frac1{1-x}\prod_{k\geq0}\frac1{1-x^{2^n}}.$$ A result of De Bruijn ...
T. Amdeberhan's user avatar
1 vote
1 answer
102 views

Integral of $J_1\left(Ae^{-\lambda t}-Ae^{-\lambda s}\right)e^{-\epsilon(t-s)}$ with respect to $s$?

Consider the integral $$\mathcal{I}=\int_0^t\left(\frac{Ae^{-\lambda t}-Ae^{-\lambda s}}{2}\right)^{2m+1}e^{-\epsilon(t-s)}ds,\tag{1}$$ for constants $A,\lambda,\epsilon,t\in\mathbb{R}$ and $m\in\...
UNOwen's user avatar
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8 votes
1 answer
352 views

Is there a real-analytic way to derive the asymptotics of $\int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$ as $|x|\to\infty$?

In The Fourier Transform of the quartic Gaussian $\exp(-Ax^4)$: Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to $\exp(-Ax^{2n})$, Boyd ...
Dispersion's user avatar
0 votes
0 answers
194 views

Subgroups generated by two random elements

Suppose that we have a finite group $G$ and choose elements $a, b \in G$ at random. What can be said about the order of the subgroup generated by $a$ and $b$? Mainly, what is the expected order, $\...
Daniel Sebald's user avatar
2 votes
1 answer
202 views

Largest asymptotic growth for $2f(x)-f(2x)$

I am trying to find smooth functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that the quantity $$\Delta_f(x) := 2f(x)-f(2x)$$ is positive for $x$ large enough and has the greatest asymptotic growth. ...
cs89's user avatar
  • 971
1 vote
0 answers
135 views

Fourier transform of the Bochner-Riesz multipliers

How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define: $$ \hat{m_{\lambda}}(x)=\int\limits_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \...
Dapao Zhang's user avatar
2 votes
0 answers
82 views

Approximate solution of nonlinear ODE

Investigating some problem in optics I am faced with a nonlinear differential equation of the form $$ - y(x)\frac{{{d^2}}}{{d{x^2}}}\left( {\frac{1}{{y(x)}}} \right) + {y^2}(x) = f(x)$$ with initial ...
Msdos4's user avatar
  • 21
3 votes
1 answer
203 views

asymptotic growth of a sum involving partitions

Let $\lambda\vdash n$ denote the integer partition of $n$. Define the product $\mathcal{N}(\lambda)=\lambda_1\lambda_2\cdots\lambda_r$ when $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r>0)...
T. Amdeberhan's user avatar
6 votes
3 answers
1k views

Is there an entropy proof for bounding a weighted sum of binomial coefficients?

Given a probability $p \in (0,1)$ and parameter $\alpha \in (0,1)$, is there an entropy-based proof which yields a good upper bound for the sum $$\sum_{\ell = 0}^{\alpha n} \binom{n}{\ell}p^\ell(1-p)^{...
Naysh's user avatar
  • 455
5 votes
1 answer
355 views

A bound for the number of integer solutions to a simple inequality

I am interested in proving an upper bound (expressed as a power of $N$, with $N\rightarrow\infty$ ) for the number of elements of the set $$ A_N=\{(k,l,m,n)\in([N,2N]\cap\mathbb{Z})^4: |k^2+l^2-m^2-n^...
Tony419's user avatar
  • 401
2 votes
1 answer
78 views

Asymptotic behavior of the moments of non-negative sequences

We consider a sequence $u = (u_k)_{k\geq 1}$ such that $u_k \geq 0$ for any $k \geq 1$. We assume that there exists a critical $p_c \in \mathbb{R}$ such that, for any $q<p_c <p$, $$\sum_{k=1}^\...
Goulifet's user avatar
  • 2,174
1 vote
0 answers
44 views

Asymptotic solution of wrinkle amplitude

I have encountered an equation determining amplitude of sinusoidal wrinkle $\int_{0}^{\lambda_0} \sqrt{1+[(A_0\sin{\frac{2\pi x}{\lambda_0}})']^2} \,dx = \int_{0}^{\lambda_0(1+\epsilon_a)} \sqrt{1+[(A\...
NamJoo Kim's user avatar
1 vote
2 answers
340 views

Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true?

According to numerical simulation, the relationship $$\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$ where $\Gamma$ is the Gamma function seems to be true. Do you ...
user avatar
3 votes
1 answer
218 views

An Euler-Mascheroni double sum

An interesting representation of the Euler-Mascheroni constant $$ \gamma~=~ \lim \limits_{n\to \infty} \sum \limits_{k,s=1}^n \frac{s-k}{k\left( s\,n +k\right)},\label{1}\tag{$*$}$$ can be proved ...
Karl Fabian's user avatar
  • 1,546
3 votes
0 answers
144 views

On an inequality for the arithmetic function counting the number of primes $\lfloor n^c\rfloor$ in the spirit of Ramanujan's prime counting inequality

In page 3 of [1] (please see if you need it the book by Berndt) Axler refers an inequality that involves the prime-counting function $\pi(x)$ and that was deduced by Ramanujan. I'm curious to know if ...
user142929's user avatar
4 votes
0 answers
94 views

Is this conjecture about the binomial and beta distributions true?

Let $X$ follow a binomial distribution with parameters $n$ and $p$, and also fix $k$ such that $1<k<n$. Define $$a = \mathbb{E}(X-k)^+$$ and $$b = \mathbb{E}\log\binom{X}{(X-k)^+}$$ where the ...
Margaret Kail's user avatar
1 vote
0 answers
44 views

Is there a local limit theorem for functions of Gaussian random vectors?

Assume that $\sqrt{n} (\boldsymbol{Z}_n - \boldsymbol{\mu}) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(\boldsymbol{0},\Sigma)$, as $n\to \infty$, for some $\boldsymbol{\mu}\in \mathbb{R}^d$ ...
Aftermath 12345's user avatar

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