Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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3
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96 views

Asymptotics of Riemann-Lebesgue type integral

How to show that for $u \in L_{\mathbb{C}}^2$ and $a>0$, $$\int_0^a u(t) \sin{\sqrt{\lambda}t} \,dt = o(e^{|Im\sqrt{\lambda}|a}),\text{ as } |\lambda| \rightarrow \infty$$ Note that $\lambda$ ...
3
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0answers
168 views

How to solve equation involving binomial coefficient?

I'm reading this paper which says If we have $$ \binom n d p^{\binom d 2} = 1 $$ where $ 0 < p \le 1$, then $$ d = 2 \log_bn - 2 \log_b \log_b n + 2 \log_b\left(\frac 1 2 e\right) + 1 + O(1) ...
3
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215 views

proving the saddle point method in a specific case

$1:$ the problem Let $f : U \to \Bbb{C}$ be analytic on some open set $U$ that includes the closed unit ball. Define a path $\gamma$ by: $\gamma(t) = e^{i t}$, -$\pi < t \leq \pi$ I want to ...
3
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218 views

Upper bound for the quality of an $abc$-triple

A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$...
2
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33 views

Asymptotic vlaue of $ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $

Inspired by this question I tried to find an asymptotic formula for $$ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $$ With the observation: $$ f(n)=\sum_{i=0}^n\frac{\lfloor \sqrt{i}\rfloor+\...
2
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25 views

How to find the asymptotic expansion of $\int_{-\infty}^{y} e^{-x^2/2}/\sqrt{2\pi} dx$ where $x \in N(0,1)$?

I realize the function inside the integral is the pdf of a normally distributed random variable x, but am unsure how to use this to solve the problem. I am trying to relate it to the inverse of the ...
2
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52 views

A Simple Stochastic Integral Asymptotics

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth condition $...
2
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30 views

Can anyone give an example of a set of numbers with arithmetic density that doesn't converge to a limit?

Question in the title. All of the examples I can think of (congruence classes, primes, etc.) converge as n goes to infinity.
2
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54 views

What's about $\sum_{n=1}^{\infty} \frac{e^{H_n}\log H_n}{n^3}$, where $H_n$ is the nth harmonic number?

I would like to do a toy verification of the Riemann hypothesis exploiting theLagarias theorem (see the section Applications in the following link) and the fact that we know a lot of decimals for ...
2
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44 views

Uniform approximation. Two boundary layers?

Find uniform approximation up to order $O(\epsilon)$: $$ \begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases} $$ At $\epsilon=0$ solutions $\pm \sqrt{1+x^2}$ don't ...
2
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36 views

Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...
2
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45 views

Estimating the number of permutations with no increasing subsequences of a prescribed length

Let $n\geq 1$ be a positive integer and let $S_n$ be the set of permutations of $\{1, \dots, n\}$ (thought of as non-repeating, exhaustive sequences of elements of $\{1, \dots, n\}$. Let $2 \leq k \...
2
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56 views

A limit about $\prod_{k=0}^\infty\frac1{1-x^k}$

If $$\sum\limits_{n = 0}^\infty {{a_n}{x^n}} = \prod\limits_{k = 0}^\infty {\frac{1}{{1 - {x^k}}}} ,$$ Prove $${a_n} < \exp \left\{ {\sqrt {\frac{{2\pi }}{3}n} } \right\}$$ and $$\mathop {\lim }\...
2
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52 views

Nested trig asymptotics

Letting $\ \ \sin^n(x)=\underbrace{\sin\circ \sin\circ\dots\circ \sin(x)}_{n\text{ times}}\ $, is it true that $\ \ \sin^n(\pi/2)\sim \sqrt{\dfrac{3}{n}}?$ More specifically, is it true that $$\...
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26 views

Why is $n \exp (-\frac{2m}{n-2}) \ge e^{-w}$?

Here $m=\frac{1}{2}n(\log n + w(n))$. The full claim is that $$\left(1-o(1)\right) n \exp \bigg(-\frac{2m}{n-2}\bigg) \ge (1-o(1)) e^{-w}$$ but am I'm having trouble seeing why. Edit: \begin{align*}...
2
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60 views

Calculus book with big/ little oh

Is there an introductory calculus textbook out there that makes good use of big/ little oh notation? Things like defining the derivative $f'(a)$ as the number such that $$f(a+\epsilon) = f(a) + f'(a)\...
2
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46 views

Asymptotic behavior of zeros of a function

Let $f(x,m)=(2m-1)\Gamma(m)\,x^{-m}$ where $x>0$ and $\Gamma(z)$ denotes the Gamma function. Let $g(x,m)=f(x,m)+f(x,-m)$. I'm interested in the solution $m=m(x)>0$ of the equation $g(x,m)=0$ ...
2
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47 views

Asymptotic behaviour of an integral depending on a parameter

I am trying to compute the asymptotics on $t$ of the following integral: \begin{equation} I(t)=\int_{\mathbb{R}^{n}}e^{-|\lambda|^{2}/2t}\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} \right)...
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23 views

higher-order (3+) Taylor expansion of a likelihood function

I was wondering what is the effect if I replace the second derivative of the log-likelihood ("Likelihood" hereafter) function with its expectation in a higher-order Taylor expansion of the likelihood ...
2
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104 views

Number-theoretic asymptotic looks false but is true?

Question Let $p_r$ be the $r'th$ prime. Is it true that, $$\sum_{r=1}^\infty s^r \ln(p_r) \sim \frac{s}{(1-s)} $$ I know this looks bizarre but kindly consider the argument below. I'm also ...
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66 views

Integral of a Gaussian with Trigonometric functions Involved

I am having a difficult time evaluating an integral unlike any integral I have seen before. To get right into things here is the integral: $$\frac{A}{\sigma_o\sqrt{2\pi}}\int_{-\infty}^\infty [\sin(...
2
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0answers
48 views

Asymptotics of recursion

suppose we have the following two sequences $$\alpha_k = (k-1)\left(1-\frac {1}{1+(k+1)l}\right) \quad , k \geq 2$$ $$\beta_k = (k-1)\left(1+\frac {1}{1+(k-1)l}\right) \quad , k \geq 2$$ where $l$...
2
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56 views

Solving for a $v$ in $\sum a_i e^{b_i (z^2+d_i) + c_i v}$

I have an equation in complex domain, $$P(e^u,e^v)=\sum_{i=1}^{N} a_i e^{b_i u + c_i v}=0 \;\;\;\text{(A)}$$ and by redefining, at the roots (I'm only showing work for one root), the first ...
2
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0answers
47 views

Is Stirling's Approximation used here, to prove the asymptotic inequalities?

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, the ...
2
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45 views

Linear convex combinations of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$, and prime counting function

Can provide us a linear convex combination of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$ a better approximation for $\pi(x)$, the prime counting function? Or not, is better $Li(x)$ ...
2
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40 views

Relation between Big O and convergence in limit of a sequence of probabilities

I'm confused on the procedure used to show the theorem 5.52 in van der Vaart "Asymptotic Statistics" p.75. Here the simplified idea. Consider the sequence of real-valued positive random variables $\{...
2
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38 views

Signs and stochastic big O notation

I'm confused on the relation between stochastic big O notation and signs. In order to illustrate my question: (1) consider a sample of i.i.d. real-valued random variables $\{X_i\}_{i}^n$, each with ...
2
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0answers
49 views

Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove (...
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23 views

Convergence in probability or convergence in distribution?

Let $X$ be a random variable. let \begin{align*} Y=\alpha_1+\alpha_2 X \end{align*} where $\alpha_1$ and $\alpha_2$ are parameters. Now let \begin{align*} Z=\hat{\alpha}_1+\hat{\alpha}_2 X \end{...
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0answers
16 views

Big $O$ notation clarification

I've encountered something of the form: $$f(n)O(g(n))$$ I think this is equivalent to $O(g(n)f(n))$, but is this true?
2
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31 views

Asymptotic behavior of the function $e^{- \lambda t^2}$ when $\lambda$ is small

I wish to prove that when $\lambda$ is taken to be very small $$ \left| e^{- \lambda t^2} - \sum_{n=0}^N \frac{(- \lambda t^2)^n}{n!} \right| = O(e^{-\frac{a}{\lambda}})$$ for some constant $a \in \...
2
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0answers
24 views

Singularity of an oscillatory integral

Given $x\in\mathbb{R}^3\setminus \{0\}$, consider the following integral: $$I(x):=\int_{\mathbb{R}^3}\frac{e^{-i|x-y|^2}}{|y|} \, dy$$ Now $I(x)$ diverges as $x$ approaches to $0$, and it seems to me ...
2
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0answers
54 views

Fastest decaying Fourier transform of a function with compact support

Square-integrable complex-valued function $y(t)$ is defined on a finite domain $t \in [0, 1]$ and satisfies the following constraint: $$ \int_0^{1} |y(t)|^2 \, dt = 1 $$ I am seeking bounds on the ...
2
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0answers
54 views

Partitioning functions into equivalence classes based on running time?

I'm studying for my midterm and doing some practice problems, and I would be grateful if someone showed how to solve this. From my understanding you have to partition the functions into equivalence ...
2
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0answers
56 views

Asymptote produced from a composition of two polynomials and an arbitrary function.

I am interested in any references and/or information regarding this sort of asymptotic relationship. Given the composition: $$\frac{f(x)h(x)+g(x)}{f(x)}$$ With the following polynomial functions $\ ...
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0answers
50 views

How to prove the following conjecture.

Prove if $d(n)$ is $O(f(n))$ and $e(n)$ is $O(g(n))$, then $d(n) + e(n)$ is $O((f(n) + g(n)))$. What I tried. if $d(n)$ is $O(f(n))$ then $d(n) \leq c_1 \cdot f(n)$ if $e(n)$ is $O(g(n))$ then $...
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0answers
59 views

Asymptotic Approximation

After analyzing performance of a cooperative system, I get the following expression for the system outage probability: $P = 1 - \frac{{{e^{ - 2\mu /{\beta _M}}}}}{{\Gamma \left( {{\alpha _3}} \right)\...
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146 views

On n! divided by a product of primes and related questions

We have the following Definition 1. For integers $n\geq 1$ we define $$f(n) = \begin{cases} 1, & \text{if $n=1$} \\[2ex] \frac{n!}{\prod_{p\leq n}p}, & \text{if $n>1$} \end{cases}$$...
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52 views

Find the angle between asymptotes

Sketch the locus of the centres of circles which touch two fixed and unequal circles. Find the angle between the asymptotes How shall I find the locus when the size of the circles are not fixed? ...
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91 views

Arithmetic progression of squarefree integers?

Let $x$ be a given positive integer. I'm intrested in the longest arithmetic progression of squarefree integers within the interval $(x,x^2)$. Both constructive and nonconstructive results. For ...
2
votes
0answers
61 views

Approximate an integral with Bessel functions

Given $r,a,\lambda\in\mathbb{R}$, $r<a$, how can I find an approximate solution for the following definite integral? $$ \int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 }}\,d\...
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votes
0answers
113 views

A modified Shapiro's Tauberian theorem? Proof or counterexample

Let $\{a(n)\}$ be a nonnegative sequence such that $$\sum_{n\leq x}a(n)[x/n^{2}]=x^{2}\log x + O(x^{2})$$ for all $x\geq 1$, where $[y]$ denotes the greatest integer $\leq y$. Is true that the ...
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0answers
44 views

asymptotics of Involutions recurrence relation

Consider the following recurrence relation where $t(n)$ is the number of involutions on $\{1,...,n\}$ \begin{equation} (n+1)t(n)+t(n+1)-t(n+2)=0 \end{equation} When $n \rightarrow \infty$, Wimp and ...
2
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0answers
56 views

Asymptotic conditional distribution of $\bar{Y}\mid\bar{X}=x$

I'm reviewing for my qualifying exam and I'm stuck on part of a problem. Setup Suppose that $(X,Y)$ are two random variables with joint distribution $ \begin{equation} f(x,y\mid\alpha,\beta)=c(\...
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44 views

An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = f(...
2
votes
0answers
59 views

WKB leading order

I'm learning about the WKB method, and I'm applying it to an assignment. The assignment question asks to find the "leading order" WKB expansion for the particular equation. For WKB you make the ...
2
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0answers
36 views

Approximation of Hermite functions

I'm looking for an "easy" proof of the asymptotic expansion of Hermite functions ($f_n(x)=H_n(x)e^{-x^2/2}$ where $H_n$ is the Hermite polynomials). The asymptotic expansion is $$ f_n(x) \sim_{n \...
2
votes
0answers
93 views

asymptotic expansion of hermite functions

Does anybody know how to proof the first asymptotic expansion of this page: http://en.wikipedia.org/wiki/Hermite_polynomials#Asymptotic_expansion ? (and how the physicist use this asymptotic ...
2
votes
0answers
23 views

Determining asymptotics of a function given a series of difference-like inequalities

I have a function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ and I know it satisfies the following properties. $f(x) \leq \frac{\log{\sqrt{2}}}{2x}$ and for all $A \geq 1$ and $B \geq ...
2
votes
0answers
43 views

Asymptotics of Laplace transform at minus infinity

I am interested in relating the asymptotic behavior of a function $f(t)$ for large values of $t$ with the asymptotic behavior of its Laplace transform $\hat{f}(s)$ for small values of $s$. In practice ...