Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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How hard must "no high-degree irreducibles" proofs be?

Let $\mathsf{RCF}$ be the usual theory of real closed fields and for $n>2$ let $\theta_n$ be the sentence "No degree-$n$ polynomial is irreducible." Since $\mathsf{RCF}$ is complete, for ...
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4 votes
0 answers
200 views

God's number for higher dimensional Rubik's cubes

In this MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every ...
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  • 3,463
2 votes
0 answers
50 views

Approximate logarithmic bound on expected maximum via central limit theorem

If $Z_i$ are standard normal, possibly dependent, one can show that $$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$ I'm looking for a similar (asymptotic) bound for asymptotically normal ...
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  • 213
3 votes
1 answer
106 views

Cycle counts in Ewens measure as $\theta$ diverges

For $w$ a permutation, let $c(w)$ denote the total number of cycles and $c_i(w)$ denote the number of $i$-cycles. The Ewens measure is a one-parameter probability distribution on permutations where ...
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3 votes
1 answer
69 views

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 ...
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  • 235
0 votes
0 answers
33 views

Problem about providing a good estimator in 2SLS

I am now studying the 2-stages least-squares method and have been curious about the following circumstances. Suppose that I have $Y_i = X^{T}_{i}β +e_{i}$ with $\mathbb{E}(e_{i}X_{i}) ̸\ne 0$, that ...
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2 votes
1 answer
191 views

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 $$dX_t = \sigma X_t \, dW_t$$...
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  • 1,149
2 votes
0 answers
134 views

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 ...
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5 votes
1 answer
124 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)\...
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1 vote
1 answer
112 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)...
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  • 529
3 votes
1 answer
111 views

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 ...
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17 votes
1 answer
546 views

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 $\varphi_p$ ...
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1 vote
0 answers
108 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 $...
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0 votes
0 answers
38 views

Bounds for ratio of Bessel functions

I know several good papers where bounds of Bessel function ration considered. For instance, the following Bessel function ratio, $$h(z) = \frac{I_{\nu}(z)}{I_{\nu+1}(z)}$$ has bounds $$h(z)<\frac{z}...
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4 votes
1 answer
199 views

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 ...
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5 votes
2 answers
144 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 ...
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  • 433
0 votes
1 answer
294 views

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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1 vote
1 answer
108 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 ...
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  • 187
0 votes
1 answer
164 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 ...
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  • 101
4 votes
0 answers
93 views

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(\...
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2 votes
1 answer
87 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 ...
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1 vote
0 answers
77 views

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 ...
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  • 11
2 votes
1 answer
173 views

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+...
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  • 2,400
2 votes
0 answers
105 views

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

Consider the nine consecutive natural numbers starting from $1584614377$. ...
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5 votes
2 answers
130 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$...
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1 vote
1 answer
123 views

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 $i\in\{1,\dots,n\}$, ...
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20 votes
1 answer
546 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{...
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  • 203
2 votes
1 answer
104 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 ...
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7 votes
2 answers
389 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 $...
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  • 1,197
5 votes
0 answers
145 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,...
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  • 111
6 votes
1 answer
183 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|<\...
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2 votes
0 answers
46 views

Asymptotic of positive solution to elliptic equation

I am reading the paper "Area minimizing hypersurfaces with isolated singularities" by Hardt and Simon (https://eudml.org/doc/152770) and I get stuck on equation 1.9 on page 106. The ...
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  • 121
2 votes
2 answers
196 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 ...
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3 votes
0 answers
68 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 ...
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  • 131
6 votes
1 answer
103 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 ...
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5 votes
1 answer
141 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 ...
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  • 141
4 votes
0 answers
158 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 ...
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1 vote
1 answer
198 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 ...
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  • 529
1 vote
0 answers
81 views

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 ...
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0 votes
0 answers
187 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 $z=0$: $$...
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3 votes
1 answer
97 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 ...
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1 vote
1 answer
89 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\...
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  • 59
8 votes
1 answer
293 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 ...
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  • 336
0 votes
0 answers
156 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, $\...
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2 votes
1 answer
181 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. ...
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  • 529
1 vote
0 answers
103 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 \...
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2 votes
0 answers
65 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 ...
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  • 21
3 votes
1 answer
179 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)...
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6 votes
3 answers
804 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)^{...
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  • 350
5 votes
1 answer
340 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^...
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