# Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

806
questions

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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 ...

4
votes

0
answers

200
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### 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 ...

2
votes

0
answers

50
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### 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 ...

3
votes

1
answer

106
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### 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 ...

3
votes

1
answer

69
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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 ...

0
votes

0
answers

33
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### 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 ...

2
votes

1
answer

191
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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
$$dX_t = \sigma X_t \, dW_t$$...

2
votes

0
answers

134
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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 ...

5
votes

1
answer

124
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### 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)\...

1
vote

1
answer

112
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### 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)...

3
votes

1
answer

111
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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 ...

17
votes

1
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546
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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 $\varphi_p$ ...

1
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0
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108
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### 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 $...

0
votes

0
answers

38
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### 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}...

4
votes

1
answer

199
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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 ...

5
votes

2
answers

144
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### 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 ...

0
votes

1
answer

294
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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 $0$ ...

1
vote

1
answer

108
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### 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 ...

0
votes

1
answer

164
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### 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 ...

4
votes

0
answers

93
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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(\...

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 ...

1
vote

0
answers

77
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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 ...

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+...

2
votes

0
answers

105
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### Average length of consecutive integers which have an increasing number of divisors

Consider the nine consecutive natural numbers starting from $1584614377$.
...

5
votes

2
answers

130
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### 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$...

1
vote

1
answer

123
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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 $i\in\{1,\dots,n\}$, ...

20
votes

1
answer

546
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### 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{...

2
votes

1
answer

104
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### 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 ...

7
votes

2
answers

389
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### 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 $...

5
votes

0
answers

145
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### 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,...

6
votes

1
answer

183
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### 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|<\...

2
votes

0
answers

46
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### 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 ...

2
votes

2
answers

196
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### 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 ...

3
votes

0
answers

68
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### 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 ...

6
votes

1
answer

103
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### 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 ...

5
votes

1
answer

141
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### 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 ...

4
votes

0
answers

158
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### 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 ...

1
vote

1
answer

198
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### 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 ...

1
vote

0
answers

81
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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 ...

0
votes

0
answers

187
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### 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 $z=0$:
$$...

3
votes

1
answer

97
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### 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 ...

1
vote

1
answer

89
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### 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\...

8
votes

1
answer

293
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### 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 ...

0
votes

0
answers

156
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### 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, $\...

2
votes

1
answer

181
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### 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.
...

1
vote

0
answers

103
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### 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 \...

2
votes

0
answers

65
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### 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 ...

3
votes

1
answer

179
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### 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)...

6
votes

3
answers

804
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### 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)^{...

5
votes

1
answer

340
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### 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^...