Questions tagged [asymptotics]

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5
votes
1answer
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
1answer
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
3answers
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
1answer
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
1answer
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
4answers
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
4answers
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
1answer
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
2answers
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
2answers
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
0answers
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
3answers
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
1answer
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
3answers
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
1answer
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
1answer
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^\...

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