# Questions tagged [asymptotics]

The asymptotics tag has no usage guidance.

639
questions

**5**

votes

**1**answer

98 views

### Why do people study Weyl asymptotics and partial-spectral-projections?

The major focus of the research that my advisor has me doing centers around the idea of asymptotic behavior of partial-spectral-projections on compact manifolds. In a few sentences, here is the ...

**7**

votes

**1**answer

139 views

### Asymptotics of sum involving Stirling numbers

I've encountered the following sum:
$$
s_n = \sum_{j=1}^n {n \brace j}(\alpha n)_j \beta^j.
$$
Here $\alpha$ and $\beta$ are positive constants, $(\alpha n)_j$ is a falling power, and ${n \brace j}$ ...

**2**

votes

**3**answers

297 views

### Is this number theoretic quantity bounded above?

I am considering a combinatorial argument which involves the following quantity. We use the prime counting function $\pi(n)$ and to save on exponents we set $h=\pi(n/2)$. The quantity as a function ...

**2**

votes

**1**answer

46 views

### Asymptotic behavior of maximum of bessel function

Let $J_n$ be the Bessel function of the first kind. Let $J_n^{(\max)} = \max_{x>0} J_n(x)$. What is known about the asymptotic behavior of $J_n^{(\max)}$ at large $n$? Specifically, I am looking ...

**0**

votes

**1**answer

77 views

### Asymptotic development of Integral of $e^xx^r$

Let $\alpha \in (0,1)$ and $\delta \in (0,1/2)$ be fixed, and consider the following integrals for each integer $j \geq 0$:
$$I_j(u):= \frac{e^u}{u^{j+\alpha}} \int_{-u\delta}^0 e^t t^{j-1+\alpha}\...

**4**

votes

**4**answers

397 views

### Can one show combinatorially how $\operatorname{lcm}(1, \dotsc, n)$ grows?

Let us write $M(n)$ for $\operatorname{lcm}(1,\dotsc,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, ...

**1**

vote

**2**answers

102 views

### Expected value of a truncated binomial

Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$. Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$? I am particularly interested in ...

**1**

vote

**1**answer

51 views

### A uniform mixture of order statistics

Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$...

**-2**

votes

**0**answers

30 views

### Is kernel density estimator a linear transformation? [closed]

I am reading the book Nonparametric econometrics, I am thinking since the kernel density estimator is given as
$$\hat{f}(x)=\frac
{1}{nh}\sum_{i=1}^nK\left(\frac{X_i-x}{h}\right),$$
where $K(\cdot)$ ...

**0**

votes

**1**answer

42 views

### Is asymptotic growth bound on a sequence equivalent to an asymptotic growth bound on its partial sum?

The following question was asked at https://mathoverflow.net/questions/360053/asymptotic-growth-bound-on-a-sequence-equivalent-to-an-asymptotic-growth-bound-o, but then deleted by the user:
I ...

**0**

votes

**1**answer

26 views

### Lyapunov condition for CLT for asymptotically independent sequence

Suppose I have some triangular array $\{X_{n,j}\}$ of random variables, which need not be independent or identically distributed. Suppose I further know that
$$Var\left(\sum_{j=1}^n X_{n,j}\right)\to \...

**7**

votes

**2**answers

572 views

### What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?

From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...

**3**

votes

**0**answers

32 views

### Are there extensions of Hilb's and Laplace's formulas to Jacobi polynomials with $\alpha,\beta\le-1$?

In Szegő's Orthogonal Polynomials book, he gives two interesting asymptotic formulas for Jacobi polynomials with $\alpha,\beta>-1$. The first (Theorem 8.21.12, page 197 is a generalization of Hilb'...

**0**

votes

**0**answers

68 views

### On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$

The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...

**1**

vote

**0**answers

59 views

### How to solve a problem from Frank Olver's book

I'm learning Frank Olver's book, called Asymptotics and Special Functions. There is an difficult exercise.
Problem. Suppose that $f,\frac{1}{f}$ possess the following asymptotic expansions :
$$f(...

**4**

votes

**1**answer

93 views

### Alternating binomial sum asymptotics

Let
$$
S_k=\sum_{j=0}^{\alpha k}(-1)^j\binom{k/2}{j}\binom{\alpha k^2-j k}{k}
$$
where $\alpha\in(0,1/2)$ is a constant.
I'm interested in understanding the asymptotic behaviour of $S_k$.
It would ...

**1**

vote

**0**answers

211 views

### Question about proof of irrationality of $\zeta(3)$ [closed]

I'm reading this article of Henri Cohen about Apery's proof of the irrationality of $\zeta(3)$ but I don't really get the details of "THEOREME 1".
My first doubt is about the relation $a_n \sim A \...

**0**

votes

**1**answer

107 views

### Local behavior of the Vandermonde convolution

An interesting combinatorial identity is the Vandermonde convolution identity:
$$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$
which can be proved by considering the coefficients in $(x+1)^...

**0**

votes

**0**answers

96 views

### Heath-Brown new form of circle method

I am following the paper A New Form of the Circle Method, and its
Application to Quadratic Forms, by Heath-Brown (see here: https://core.ac.uk/download/pdf/96603.pdf). I would like some help ...

**0**

votes

**0**answers

99 views

### Finding a square integrable dominating function for function class

problem statement
For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$
where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...

**3**

votes

**0**answers

50 views

### Limit of Hankel function for large complex order, fixed real argument

Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...

**2**

votes

**1**answer

171 views

### Seeking a Lyapunov function for a SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\...

**0**

votes

**1**answer

89 views

### How to check if you have the asymptotic solution of some equation? [closed]

Suppose I have an analytic function $f : \mathbb{R} \to \mathbb{R}$ and I have the asymptotic expansion of some $x_0$ up to a few terms in terms of $\epsilon$ for some $\epsilon \to 0$ which I believe ...

**3**

votes

**2**answers

114 views

### Meinardus theorem at use: problems with conditions

I am working on an enumerative problem related to knot theory, and I have found the following generating function
$$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$
I am interested on getting ...

**1**

vote

**0**answers

54 views

### Spitzer's condition, a slowly varying function and its behavior

Let $S$ denote a random walk that satisfies Spitzer's condition $$ \frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, ...

**8**

votes

**4**answers

635 views

### Estimate for $\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}\frac{b}{a(ab)_p}$, where $p$ is a large prime

Is this estimate true? Can anyone give a proof of it?
$$
\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}\frac{b}{a(ab)_p}=\frac{1}{2}p\ln^2 p+o(p\ln^2 p)\qquad (p\text{ prime, } p\to\infty)
$$
where $
(ab)_p\equiv ...

**6**

votes

**1**answer

82 views

### Stationary phase in spherical integral

I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that:
If $\lambda\gg 1$...

**2**

votes

**0**answers

160 views

### Asymptotic of a functional as $x\rightarrow \infty$

Consider the following functional :
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},
$$
where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...

**2**

votes

**2**answers

94 views

### Convergence of fraction of expectation values

Let $X_1,...,X_n$ be iid normal random variables.
I am looking for a strategy to establish the following limit for fraction of expectation values
$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i ...

**2**

votes

**1**answer

87 views

### Convergence of estimator given by a fixed point

Let $X$ be a non-negative random variable with cdf $F$ and define
$$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function.
Let $s_0$ be the unique fixed point of $G$.
Now let $X_1,\dots,X_t$ ...

**1**

vote

**0**answers

77 views

### Minima of a random walk and an equality for a fraction

Let $S_n := X_1 + \dots + X_n$ denote a random walk with zero mean and finite variance and write $L_n := \min \{ 0, S_1, \dots, S_n\}$. The tail distribution of $L_n$ are well-known and in particular,
...

**1**

vote

**2**answers

255 views

### Simple bound on $\log(x)/x$

I would like to pick $x$ as small as possible while guaranteeing that $\log(x)/x \leq \epsilon$ where $\epsilon \ll 1$. Clearly $x$ should (roughly) be of the order $1/\epsilon$; I would like simple ...

**2**

votes

**1**answer

224 views

### Asymptotic of an area integral

I have the following integral
$$
I(\varepsilon) = \iint_D \frac{\sqrt{1+|\nabla h(u,v)|^2}}{[(h(u,v)+\varepsilon)^2+u^2+v^2]^2} du dv,
$$
where $h$ is a smooth function with
$h(0,0)=0 = h_u(0,0) = h_v(...

**2**

votes

**0**answers

186 views

### Asymptotic of an integral

Let
\begin{equation*}
V(x) = -\big(2-\sin(2\pi x) - \sin(2\pi \sqrt{2}x)\big)^\gamma
\end{equation*}
for some $\gamma \in (0,1]$. Define for each $r<0$ the number
$$a_r = \min\{a>0: V(a) = ...

**0**

votes

**1**answer

36 views

### Deriving asymptotic variance of generalized estimating equation estimator (GEE)

As well known to us, K.Y. Liang and S. Zeger proposed GEE for longitudinal data analysis in their famous paper[1]. At the appendix of the paper, authors show the proof of Theorem 2. I tried to ...

**1**

vote

**0**answers

156 views

### Asymptotic expansion for sum involving divisor function [closed]

Could someone tell me if there is a more precise asymptotic expansion for this sum?
$$\sum_{n\leq x}\frac{d(n)}{n}=\frac{1}{2} \ln^{2}x+2\gamma \ln x+c+O(x^{-\frac{1}{2}}\ln x) $$
$$d(n)=\sum_{a|n}1$$...

**3**

votes

**0**answers

33 views

### Proving that a quotient of hypergeometric functions is smaller than a certain function

Im trying to prove that $\forall w \in (0,1), \forall k \in \left(0,\frac{1}{5}\right)$:
$$h_k(w) = \left[\frac{_2F_1\left(\frac{3}{2},1+\frac{1}{k};\frac{1}{2}+\frac{1}{k};\frac{1-w}{1+w} \right) }{...

**-2**

votes

**1**answer

133 views

### Asymptotics for certain integrals

I stumbled on the following problem, if you can see a way through it.
Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$.
QUESTION. For $x\rightarrow0$, does there exist a ...

**0**

votes

**0**answers

26 views

### Bound for $\left|\sum_{\substack{1\leq d\leq x\\(d,r)=1}}\frac{G_d}{d^{1+\varepsilon}}\right|$, where $G_d$ denote the Gregory coefficients

In this post we denote the Gregory coefficients, or reciprocal logarithmic numbers, this Wikipedia Gregory coefficients as $G_k$, for integers $k\geq 1$. I would like to know if it is possible to get ...

**2**

votes

**2**answers

178 views

### Asymptotic decay rate of an oscillatory integral

Consider the following oscillatory integral
$$
I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac
{(1 - \cos(2x)) (1 - \cos(2y))}
{2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.
$$
where $...

**2**

votes

**0**answers

92 views

### Real root of the derivative of a prime cyclotomic polynomial

Consider the graph of $y=x^n$ with $n$ odd, and draw a tangent to its negative arc that crosses the graph at $(1,1)$. Equating the slopes gives us $\frac{x^n-1}{x-1}=nx^{n-1}$ at the point of tangency....

**0**

votes

**1**answer

60 views

### Asymptotic expansion of hypergeometric 2F2

I would like to find an asymptotic expansion for the hypergeometric function
$$
_{2}F_{2}\left(a,b;c,d;z\right),\quad a,b,c,d\in\mathbb{R}.
$$
The parameters are fixed. $z$ is real and $z\rightarrow ...

**0**

votes

**2**answers

92 views

### The exact constant in the simple bound of the fraction of Gamma Functions

In the Question : Upper bound of the fraction of gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...

**3**

votes

**2**answers

110 views

### Trying to bound the generalized hypergeometric function ${}_2F_3(x+1,x+1;1,1,1;\alpha)$ as $x\to \infty$?

(See also edit below)...
I am trying to get a nice, explicit, bound on the hypergeometric function
$$
{}_2F_3(a_1,a_2;b_1,b_2,b_3;\alpha),
$$
in the case of a large parameter. In particular I am ...

**1**

vote

**0**answers

72 views

### Asymptotic behaviour of $\sum_{k=1}^{n}\frac{R_{k+1}+R_k}{R_{k+1}-R_k}$, where $R_k$ are the Ramanujan primes

I was inspired in the following post from this Math Overflow that was asked yerterday Asymptotic behavior of a certain sum of ratios of consecutives primes to ask about a similar question for ratios ...

**16**

votes

**3**answers

604 views

### Plane partitions with equal margins

A plane partition of $n$ is an table of integers $A=(a_{ij})$ which add up to $n$ and non-increase in rows and columns. For example,
$$A= \begin{matrix} 331 \\
32 \ \ \\
11 \ \
\end{matrix}
$$
is a ...

**2**

votes

**1**answer

138 views

### Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution

$\mu=1+\epsilon$ where $\epsilon>0$ holds.
1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$
This quantity can be ...

**3**

votes

**3**answers

157 views

### Reconstructing probability distribution with high probability

Sample $m$ times from unknown probability distribution $p=(p_1,p_2,\cdots,p_n)$, we can construct a probability distribution $q=(q_1.q_2,\cdots,q_n)$.
How large $m$ should be to achieve that the ...

**2**

votes

**1**answer

118 views

### Tight sublinear estimates for a triple partial binomial summation

Is there tight estimates for the following logarithmic summation ($\gamma,\gamma'\in(0,1)$ and $\mu,\mu'>0$)
$$\log_2\Bigg(\sum_{t=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{...

**1**

vote

**1**answer

97 views

### Tight estimates for binomial summation

Is there tight estimates for the following logarithmic summation ($\gamma\in(0,1)$)
$$\ln\Bigg(\sum_{t=\frac{n^{}}2-\gamma n^\gamma}^{\frac{n^{}}2+\gamma n^\gamma}\sum_{\ell=\frac{n^{}}2-\gamma n^\...