Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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Sum of reciprocal of Pochhamer symbols through multiples of a natural L

In a question in StackExchange (https://math.stackexchange.com/questions/4236635/sum-of-quotients-of-gamma-functions), I asked if there is a closed expression for the following sum related with a ...
George McGonagall's user avatar
1 vote
1 answer
418 views

Asymptotic cone

Let $S$ be a subset in a real vector space $\mathbb{R}^n$. Define the asymptotic cone $S\infty:=\{y\in\mathbb{R}^n\mid\textrm{there exists a sequence }(y_k,\varepsilon_k)\in S\times\mathbb{R}^+\textrm{...
Hebe's user avatar
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Probabilistic behavior of greedy point selection in the plane

Let $\mathcal{X} = X_1,\dots,X_n$ be a collection of independent, uniform samples in the unit square. Let $\mathcal{S}=\{X_1\}$, and consider the following process: for $i=2,\dots,n$, let $x^*$ be ...
Tom Solberg's user avatar
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6 votes
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Asymptotic estimate for an integral involving the squared modulus of the Riemann zeta function

For any fixed $\frac{1}{2} < \sigma < 1$, let $$\int_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \ dt = O(T^\theta), \qquad T \uparrow \infty. $$ It is clear that $\theta > 0$, since we ...
nickkatzfl's user avatar
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154 views

On analogues of Weber's formula

Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that $$ \int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta})....
Alexander Kalmynin's user avatar
4 votes
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112 views

Asymptotics of holonomic recurrences and the Birkhoff-Trjitzinsky method

While reading about asymptotics of holonomic recurrences, I had the following impressions that are related to the divulgation of the theory: I don't know what is the current status of the divulgation ...
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Finding tighter upper bound of a complicated function

Define the following functions from $[0,1) \to \mathbb{R}$: Let $f(x)=1-x^n$. Let $r=1-\frac{1}{n(1-x)+1}$ and $u(x)=\sum_{i=1}^n\binom{n}{i}\frac{x^{n-i}(1-x)^i(1-r^i)}{(1-x^n)(1-r)i}$. Let $g(x)=1-\...
Maxim Enis's user avatar
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About the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$

Let the sequence: $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$, for $mn>0$. Computationally it seems that $\frac{s_{n+1}}{s_{n}} \approx e\cdot ...
José María Grau Ribas's user avatar
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Asymptotics of $\sum \frac{d(n)}{n}$ with generating functions

We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\...
Max Muller's user avatar
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Asymptotic behavior of hypergeometric function ${}_{3}F_{2}(a,b-n,c-n;d-n,e-n;1)$ for $n\to\infty$

Suppose $a,b,c,d,e\in\mathbb{R}$ are such that $d+e-a-b-c>0$ and $d,e\notin\mathbb{Z}$. I would like to know whether it is possible to deduce an asymptotic formula for the sequence given by the ...
Twi's user avatar
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Approximation of $\Phi (p)$

I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe ...
user311932's user avatar
2 votes
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Asymptotic bound of quotient of absolute and squared deviation from mean

The following fraction shows up when trying to show consistency of the OLS slope estimator in a simple linear regression on a log-log scale where the window of observation changes as the sample size $...
AlbertRapp's user avatar
1 vote
1 answer
116 views

Asymptotics of the integral of an oscillating function

I would like to know the asymptotics of the following sequences of integrals: $$ I_n = \displaystyle { \int _0 ^{+ \infty} \dfrac{t^n}{(t + i)^{n + 1}} ...
MathTolliob's user avatar
2 votes
0 answers
71 views

Uniform bound on a certain family of hypergeometric functions

We have the following problem, which we can't solve. Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
Sasha's user avatar
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13 votes
2 answers
435 views

Asymptotics of a randomized Fibonacci sequence

Let $f(1)=f(2)=1$ and recursively define $f(n+1) = f(n) + f(i)$, where $i$ is chosen uniformly at random from $1,\ldots,n-1$. About how big should we expect $f(n)$ to be for $n$ large? We can examine ...
Christopher D. Long's user avatar
3 votes
2 answers
133 views

Asymptotics of a sequences of integrals

I would like to know the asymptotics of the following sequences of integrals: $$ I_n = \int _0 ^{+ \infty} e^{-t} \left ( \dfrac{t}{1 + t} \right )^n \...
MathTolliob's user avatar
6 votes
1 answer
202 views

Reference request: Different definitions of Big O notation

This question might sound strange, but I would like to settle this problem once and for all. For as long as I can remember, I was introduced to the Big O notation by this definition: Def. 1: Let $f, g$...
MegV's user avatar
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1 answer
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Asymptotic behavior of an ODE

Consider the following ODE eigenproblem of $y(x)$ \begin{equation} y'' + [\varepsilon + b^2 x - (a + \frac{b^2}{2}x^2)^2 ] y=0 \end{equation} with eigenvalue $\varepsilon$, real constants $a,b$. ...
xiaohuamao's user avatar
3 votes
1 answer
256 views

Is there an asymptotic bound between converging and diverging series? [closed]

Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$, $$ \log^{[k]}(x) = \begin{cases} \log^{[k-1]}(\log(x)) & k>0 \\ x & k=0 \end{cases}. $$ It is well known, ...
Niv Sarig's user avatar
2 votes
2 answers
133 views

Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type

Let $J_\nu$ be a Bessel function of the first kind and let $\{\lambda_{n, \nu}\}_{n\ge 1}$ be a sequence of its zeroes. I claim that $$ \inf_{n\ge 1}\bigg|\sqrt{\lambda_{n,\nu}} J_{\nu+1}(\lambda_{n,\...
Tony419's user avatar
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Will an integer combination of some number of copies of the set of powers of 2 and the set of powers of 3 always have natural density 0?

Consider a Diophantine equation of the form $$(c_1 2^{x_1} + \dots + c_n 2^{x_n}) + (c_{n+1} 3^{x_{n+1}} + \dots + c_m 3^{x_m}) = y$$ where $x_1, \dots, x_m, y$ are our variables (here $x_1, \dots, ...
Christian Schulz's user avatar
4 votes
1 answer
95 views

Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$

For every $x,y\in\mathbb R$ let $$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$ where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})...
tituf's user avatar
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Existence of function $f$ such that $f(x) \sim \sum_{j \in \mathbb{N}} x^{1 - \frac{1}{j}}$

Is there a function $f$ on $\mathbb{R}$ such that as $x \to 0$, $$ f(x) = \sum_{j=0}^N x^{1 - \frac{1}{j}} + o(x^{1- \frac{1}{N}}), $$ for every $N \in \mathbb{N}$? Heuristically there shouldn't be ...
Jannik Pitt's user avatar
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0 answers
94 views

Asymptotic optimal sphericity

How quickly does maximum sphericity of polyhedra with $n$ faces approach 1 as $n→∞$? I can show that sphericity $1 - \frac{5 \sqrt{3} π}{27n} - O(n^{-3/2})$ is possible. Is this, especially $O(n^{-3/...
Dmytro Taranovsky's user avatar
9 votes
0 answers
356 views

Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$

Consider the generating function of "partitions with distinct parts" $$\sum_nQ(n)x^n=\prod_k(1+x^k).$$ It's known that $$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...
T. Amdeberhan's user avatar
3 votes
0 answers
114 views

Asymptotics of a combinatorial series

I am interested in the exact asymptotics of the following combinatorial series (which arises from the study of a Markov chain): $$F(q):=\sum_{k \ge 1} \frac{q^{k^2}}{(q;q)_k^2}\quad \mbox{as } q \to 1^...
KDD's user avatar
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4 votes
2 answers
263 views

Using $\delta$-method to "estimate" undefined moments of a random variable?

I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is. Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...
Aaron Hendrickson's user avatar
5 votes
1 answer
594 views

Is the harmonic series worse than any summable series?

It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values. We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \...
Sascha's user avatar
  • 506
1 vote
1 answer
85 views

Asymptotics of the right singular vectors as the number of rows diverge [duplicate]

Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...
user257566's user avatar
4 votes
0 answers
198 views

Product of all multinomial coefficients

At some moment I found that for computing a bound on a density, I need to compute (or find a good asymptotic) of the product of all multinomial coefficients, i.e., $$ \prod_{\substack{(\alpha_1,\ldots,...
Josué Tonelli-Cueto's user avatar
3 votes
1 answer
143 views

Randomized version of Turán's theorem II

$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then \begin{equation} \om(G)\ge\...
Iosif Pinelis's user avatar
5 votes
1 answer
206 views

Randomized version of Turán's theorem

Turán's theorem says the following. Take any natural $n$ and $r$. Suppose that \begin{equation*} |G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0} \end{equation*} where $|G|$ is the number of edges of ...
Iosif Pinelis's user avatar
1 vote
1 answer
167 views

Ratio limit results for restricted partition functions

This concerns difference/limit ratio results for special restricted partitions. Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $...
David Handelman's user avatar
4 votes
1 answer
182 views

Is there an elementary subexponential upper bound on the size of the stable stems?

This is a question in stable homotopy theory which I will boil down to a pure combinatorics question. If you're not interested in the homotopy theory, feel free to skip to the end for the ...
Tim Campion's user avatar
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0 votes
1 answer
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Perturbative approach starting from a probability distribution approximated form

I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$, such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity. Consider the generic ...
user1172131's user avatar
1 vote
1 answer
89 views

Critical point of saddle point equation

Consider the following integral: \begin{equation} \int \mathrm{d}\rho \frac{1}{\rho} e^{N f(\rho)} \end{equation} Where: \begin{equation} f(\rho)=\ln \rho-\frac{1}{2} \rho^{2}+\frac{1}{2 p w^{2}} \rho^...
Matt's user avatar
  • 97
9 votes
3 answers
691 views

Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$

Problem: Let $x_1 = 1$ and $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2, \ n\ge 1$. Find the third term in the asymptotic expansion of $x_n$. I have posted it in MSE six months ago without ...
River Li's user avatar
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2 votes
0 answers
76 views

Entropy per site of quantum spin chain

It’s fork lore that von Neumann entropy (and free energy) grows linearly with respect to the size of a quantum system. Is there a rigorous demonstration in the toy model of a quantum spin chain with (...
Xu Kai's user avatar
  • 189
2 votes
2 answers
303 views

Asymptotic of an improper integral

I found myself stuck with an "elementary" claim in some article. A simplified version of the problem is: Let $p : [0,1]\to [0,1]$ be a continuous and non decreasing function such that $p(0)=...
Chev's user avatar
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1 vote
0 answers
75 views

Approximating the partial sum of remainders function

This is a question related to the one I posted here, but I have found some more interesting and general results and thought here might be a better place to ask. Let $R_{k,N}$ denote the remainder of ...
Fred Li's user avatar
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0 votes
0 answers
115 views

Can the Bessel functions tend to a plane wave?

Can the Bessel functions tend to a plane wave? If I have this function: $$ y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6) $$ ...
Gin's user avatar
  • 23
9 votes
1 answer
548 views

Two-term recurrence relation

We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$ $$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$ ...
Kung Yao's user avatar
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2 votes
0 answers
125 views

Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

I have a system of nonlinear Volterra integral equations of form $$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$ and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
Quiet_waters's user avatar
0 votes
1 answer
173 views

Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function

Let's denote $F_{t_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t_u$ with $u$ degrees of freedom and $F_{t_v}(x)$ the cumulative distribution function of the t-distribution $t_v$...
NN2's user avatar
  • 250
1 vote
0 answers
92 views

Convergence result on Cornish Fisher expansion of binomial distribution

Since it is known that Cornish Fisher expansion of quantiles does not have guaranteed convergence for all distribution, I wonder specifically if any convergence result is known in literature for CF ...
messi22's user avatar
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2 votes
0 answers
323 views

For the following class of matrices, are the determinants invariant under permutations?

I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
Charles Cao's user avatar
0 votes
1 answer
176 views

lower bound for sum of the n factors of the inclusion exclusion principle

Suppose the following relation is established: $P\Bigl(A\cup B \cup C\cup D\Bigr) < P\Bigl(E\cup F\cup G\Bigr)$ based on boole's inequality, for each of the above probabilities we can have the ...
Optimized Life's user avatar
1 vote
1 answer
96 views

Tail bounds on random series in Hilbert space

Tail bounds on random series in Hilbert space Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$, $n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ ...
Yilmis's user avatar
  • 11
5 votes
1 answer
242 views

Asymptotics of ratios of polynomially recursive sequences

A sequence $a_n$ is said to be polynomially recursive (P-recursive) if it satisfies: $$p^{[r]}(n)a_{n+r}+\cdots+p^{[1]}(n)a_{n+1}+\cdots + p^{[0]}(n)a_n=0$$ where $p^{[i]}(t)\in \mathbb{Q}[t]$ are ...
Alex R.'s user avatar
  • 4,902
6 votes
1 answer
341 views

Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$

To begin, let us set $$A_Q(n):=\sum_{d|n \\ d<Q}\mu(d)$$ If we fix $Q$ and let $n$ vary, we get a very surprising amount of cancellation. For instance, the trivial bound \begin{align*} \mathbb{E}_{...
Milo Moses's user avatar
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