Skip to main content

Questions tagged [asymptotics]

For questions involving asymptotic analysis, including function growth, Big-$O$, Big-$\Omega$ and Big-$\Theta$ notations.

Filter by
Sorted by
Tagged with
-1 votes
0 answers
20 views

Find all $\alpha, \beta, \gamma$ such that $(C_{2^n}^n)^{\frac1n}\sim\alpha n^{\beta} \gamma^{n}$

I'd like to get some help in solution of the following task: Find all real $\alpha, \beta, \gamma$ such that $(C_{2^n}^n)^{\frac1n}\sim\alpha n^{\beta} \gamma^{n}$ I've found that the left part is ...
John Doe's user avatar
0 votes
0 answers
10 views

Finding constant for asymptotic equality

I have the following task and I'd like you to check my thoughts on it: Is there a constant C such that when $n \to \infty$ it is true that $\frac{(2n)!}{n!n^n} = (C + o(1))^n$ If such constant C ...
Jane Doe's user avatar
0 votes
0 answers
18 views

Consequence related to Bollobas' theorem about chromatic number of a random graph

I've came across the following theorem: let $G(n,p)$ be a random graph and let $p = p(n) = n^{-\alpha}, \alpha \in (\frac{1}{2}, 1)$ then there is a function $u=u(n, \alpha)$ such that asymptotically ...
Jane Doe's user avatar
0 votes
0 answers
13 views

Convergence in $𝐿^1$ and continuos CDF , implies converges in $𝐿^1$ for the corresponding binarized succession?

Let $X_n$ a succession of random variables with $\{x \in \mathbb{R}: \mathbb{P}(X_n=x)>0\} \subseteq[0,1]$ $\forall n$, and X a random variable with $\{x \in \mathbb{R}: \mathbb{P}(X=x)>0\} \...
Riccardo Cadei's user avatar
2 votes
0 answers
40 views

Asymptotic expansion of a differential equation

I am a physicist studying a nonhomogenous nonlinear differential equation of the general form \begin{equation} a(x,f,f^\prime)f^{\prime \prime}(x) +b(x,f)f^\prime(x) +c(x)f(x) = \mathcal{F}(f(x),x), \...
Bairrao's user avatar
  • 121
0 votes
0 answers
10 views

On little-o notation and exponent

I need to get asymptotic for $(3+\frac{1}{\sqrt{n}})^n$ My attempt is: $(3+\frac{1}{\sqrt{n}})^n = 3^n(1 + \frac{1}{3\sqrt{n}})^n = 3^n((1 + \frac{1}{3\sqrt{n}})^{3\sqrt{n}})^{\frac{\sqrt{n}}{3}}=3^ne^...
Jane Doe's user avatar
0 votes
0 answers
10 views

How to find $\delta$ such that the formula growth with n [closed]

How to write the lowest $\delta(n)\in(0,1)$ we can get such that $$\frac{n^2\delta^2(1-2p)^2}{p(1-p)}$$ tends to infinity when n tends to infinity? Here $p$ is a function on $n$ that approaches $1/2$ ...
chloe's user avatar
  • 554
-1 votes
1 answer
49 views

Convergence in $L^1$, implies converges in $L^1$ for the corresponding binarized succession?

Let $X_n$ a succession of random variables with $\{x \in \mathbb{R}: \mathbb{P}(X_n=x)>0\} \subseteq[0,1]$ $\forall n$, and X a random variable with $\{x \in \mathbb{R}: \mathbb{P}(X=x)>0\} \...
Riccardo Cadei's user avatar
1 vote
0 answers
27 views

How to tell which of several possible asymptotic forms a numerical solution to an ODE is converging to

I've previously mentioned the ordinary differential equation $$12P\left(f\left(x\right)\right)^3f''''\left(x\right)+12\left(3P-1\right)\left(f\left(x\right)\right)^2f'\left(x\right)f'''\left(x\right)+...
Daniel Hatton's user avatar
0 votes
1 answer
32 views

One counterexample of $f\sim g\nRightarrow 3^f=\Theta(3^g)$

At first glance, $$f\sim g\Rightarrow \lim_{x\to \infty}f/g=1\Rightarrow \lim_{x\to \infty}c^f/c^g=1,c\text{ is one constant}\Rightarrow c^f\sim c^g\Rightarrow c^f=\Theta(c^g).$$ But my textbook says ...
An5Drama's user avatar
  • 346
0 votes
2 answers
77 views

Asymptotic expansion / behaviour of integral function at large x [closed]

How can I find the asymptotic expansion of the following integral, i.e. its behavior for large $x$? $$ \int_0^x \frac{y^k}{\sqrt{1+y^l}}dy $$ I know that the integral can be solved exactly for certain ...
olse barn's user avatar
0 votes
0 answers
15 views

Equivalent estimation of ARMAX models of time series

Suppose I have an ARMAX model as follows: \begin{equation}\label{eq:ARMAX} \begin{split} \ln{\epsilon_t}^2=\phi_0+&\sum_{i=1}^{p_1} \phi_i \ln{\epsilon_{t-i}^2}+\sum_{j=1}^{p_2} \phi_{p_1+j} ...
entropy's user avatar
  • 147
0 votes
1 answer
40 views

Function that grows asymptotically as $x\log(x)$ or $\log(x!)$, but is even? Function that grows as $\log(x)$ but is odd?

I'm trying to implement a model for predicting origin location - destination location traffic, from road traffic (if you are interested in the model, it is from this paper: Bell1983). The author ...
me9hanics's user avatar
  • 103
2 votes
2 answers
90 views

Prove that $\log_2(n) \le \sqrt{2n}$ for all $n\ge1$

I have proven by induction that for every $k\ge4$, $\log_2(2^k)\le 2^{k/2}$. I need to prove that $\log_2(n)\le \sqrt{2n}$ for all $n\ge1$ by using the fact that $\log_2(2^k)$ for the smallest integer ...
serendipity0217's user avatar
-1 votes
0 answers
43 views

Bounding the sum

Is the following statement true? If so, how to prove it? If not, what is a counterexample, and are there any mild conditions or slight modifications that can make it correct? Given any sequence of ...
Daniel Mendoza's user avatar
3 votes
5 answers
146 views

Asymptotic behavior of $\int_{0}^{\infty} \frac{\cos(x)}{\sqrt{x+r}} dx$ as $r \to \infty$

Sometimes I come across integrals that seem like they should have a nice asymptotic expansion, but I get nervous when the terms of the expansion seem to fail to converge. The following is an example ...
user196574's user avatar
  • 1,740
1 vote
1 answer
50 views

Prove functions equivalence

How can we show that the following notations are equivalent? Let $f:\mathbb{N} \to \mathbb{R}$ if this function can be written as $(C + o(1))^n, C > 1 \Leftrightarrow \ln{f(n)} \thicksim n\ln{C}$ I ...
Jane Doe's user avatar
0 votes
0 answers
27 views

Asymptotic of number of edges in a graph without k-cliques

I need to find asymptotic of number of edges in a graph on $n$ vertices without $k$ cliques (without $K_k$ subgraphs). I need to to it using Turan's theorem. Here is my attempt: let $\alpha(G) = k - 1$...
Jane Doe's user avatar
12 votes
1 answer
719 views

Taking the inverse (not the reciprocal) of both sides of an inequality

This is something I'm having a hard time finding online, but say we know that $f(x) > g(x)$ (for all inputs $x > a_{0}$ for some $a_{0}$), then would it always be true that $f^{-1}(x) < g^{-1}...
Bob Marley's user avatar
5 votes
1 answer
166 views
+50

The asymptotic of an integral $I$

Consider the integral $$ I(\lambda)=\int_0^1 \frac{1}{\sqrt{v}}\,\left( \int_{-\infty}^{+\infty} \frac{e^{i\lambda u (u^2-v)}}{\sqrt{u^2+ 4v}}\,\varphi(u,v)\, du\right) dv, $$ where $\varphi\in C_0^\...
cbi's user avatar
  • 107
0 votes
0 answers
38 views

Big O notation problem (dividing two big o notations)

As in topic I got problem with understanding exercise. Task is to check if this is true or false. For me it's true, but my teacher said that this is false but I don't know why. Can anyone explain it ...
trulion's user avatar
  • 11
0 votes
1 answer
58 views

If $\sum_{p<n} f(p)\sim n$ then what can be said about $\sum_{p<n} f(p)/p$?

Setup: Let $f:\mathbb{Z}_{>0}\to\mathbb{C}$ be an arithmetic function for which $\sum_{p<n} f(p)\sim n$. Question: What can be said about the asymptotic behavior of the sum $\sum_{p<n} f(p)/p$...
Tristan Phillips's user avatar
-1 votes
0 answers
42 views

Asymptotic analysis of $n^{\log n}$ vs $(\log n)^n$ [closed]

I want to find the asymptotic relationship between the functions $$\begin{align*} f(n) &= n^{\log n} \\ g(n) &= (\log n)^n. \end{align*}$$ I can see from a Desmos plot that $\lim_{n \to \infty}...
jim's user avatar
  • 1
0 votes
0 answers
22 views

Asymptotic analysis of $(\log n)^{\log n}$ vs $n^{\log \log n}$

What is the most informative relationship between the functions $$\begin{align*} f(n) &= (\log n) ^ {\log n} \\ g(n) &= n ^ {\log \log n} \end{align*}$$ out of $f = O(g)$, $f = \Omega (g)$, $f ...
jim's user avatar
  • 1
1 vote
0 answers
27 views

About the behaviour of an integral for $|x| > 1$ and $|x| < 1$

Let $f = \chi_{B(0,1)}$. Can anyone help me with the behavior of the following convolution $$f * |\cdot|^{-\alpha}(x) = \int_{B(0,1)}\frac{1}{|x-y|^{\alpha}}dy,$$for the cases $|x| > 1$ and $|x| &...
user57's user avatar
  • 760
0 votes
0 answers
19 views

Why contour with constant imaginary part implies rapid change in real part

In the method of steepest descent, considering integrals of the type $$J(z)=\int_{C}e^{zf(t)}dt$$ where contour $C$ is such that the integrand goes to zeros at the ends of the contour, the method ...
user135626's user avatar
  • 1,309
0 votes
1 answer
34 views

Stochastic Op bound for max of set of stochastic processes.

Let $X = \{X_1, \dots, X_p \}$ be a set of stochastic processes such that $X_i = O_p(a_n) \; \forall i$. Is it true that $\max{X} = O_p(a_n)$ ? My attempt: Let $\epsilon > 0$ we know that there ...
Dylan Dijk's user avatar
0 votes
1 answer
66 views

If the sum $\sum_{n=1}^\infty n a_n$ converges for positive $a_n$, what can we say about the sequence $a_n$? [closed]

Let $(a_n)$ be a sequence of positive numbers and assume that $$\sum_{n=1}^\infty n a_n < \infty.$$ What can we then say about $a_n$? There are a few obvious things; $a_n\to 0$ and the series $\...
Snildt's user avatar
  • 354
2 votes
1 answer
49 views

Multiderangement probability goes to $e^{-k}$

It's well known that the probability of a $n$-permutation being a derangement is about $e^{-1}$ when $n$ is large. But how about if we have $k$ of each number, i.e. we have multipermutations of $M_{k,...
ploosu2's user avatar
  • 9,216
0 votes
1 answer
34 views

Application of the Method of Steepest Descents to Exponential Integral

I am trying to develop an asymptotic expansion of the following integral using the method of steepest descents: $$ \int_{0}^{\infty} \frac{1}{t+1}e^{ix(t^3-3)}dt$$ I rearranged it into the form $\int ...
S M's user avatar
  • 71
3 votes
1 answer
50 views

Conjecture: The $n$th left Riemann sum for $\int_0^1 (x-x^2)^k dx$ is $B(k+1,k+1) + \Theta(n^{-2 \lceil (k+1)/2\rceil})$

While playing around on Wolfram Alpha with integrals of the form $\int_0^1 (x-x^2)^k dx$, where it happens that Wolfram Alpha displays formulas for the $n$th left Riemann sum (using equally spaced ...
Daniel Schepler's user avatar
1 vote
3 answers
58 views

Does $f(x + o(1/n)) = f(x) +o(1/n)$ holds if $f$ is a real-valued continuous function?

If $f:\mathbb{R}^n \to \mathbb{R}$ is a smooth function, then we may invoke Taylor's expansion and claim that $f(x+o(1/n)) \approx f(x) + \nabla f_x^T o(1/n) $, if $\nabla f$ is bounded, then we have $...
zt wang's user avatar
  • 363
0 votes
0 answers
63 views

Looking for a solution of $\sum_{i = 1}^{k} \sum_{{d}_{1}\, {d}_{2} = i (2k - i), {d}_{1} \le N, {d}_{2} \le N} [GCD(2 k, {d}_{1}, {d}_{2}) = 1]$

The double sum is $$\sum_{i = 1}^{k} \sum_{\substack{{d}_{1}\, {d}_{2} = i \left({2k - i}\right), \\ {d}_{1} \le N, {d}_{2} \le N}} \left[{\left({2\, k, {d}_{1}, {d}_{2}}\right) = 1}\right]$$ where [.....
Lorenz H Menke's user avatar
3 votes
2 answers
426 views

A question on Gamma function

This might be basic but I have difficulty understanding what exactly goes wrong in the following logic: Consider the Gammma function $$\Gamma(z) = \int_0^{\infty} t^{z-1} \, e^{-t}\,dt \quad \textrm{...
Ali's user avatar
  • 368
1 vote
1 answer
61 views

Asymptotic description giving a general result on interval coloring

Before I ask my question, let me give some definitions: Definition(Interval Coloring): Let $G$ be a loopless graph. Then, a (proper) edge coloring of $G$ with $t$ colors is called an interval $t$-...
ArsenBerk's user avatar
  • 13.3k
0 votes
1 answer
45 views

Lower bound for the prime zeta function

The prime zeta function is defined for $\mathfrak{R}(s)>1$ as $P(s)=\sum_{p} \ p^{-s}$, where $p \in \mathbb{P}$. It is well-know this series converges whenever $\mathfrak{R}(s)>1$. Now, ...
Frank Vega's user avatar
0 votes
1 answer
107 views

Notation higher order terms in Taylor expansion

Let's consider the Taylor expansion of $\sqrt{1-x}$, which is given by: $$\sqrt{1-x} = 1-\frac{x}{2} - \frac{x^2}{8} + \mathcal{O}(x^3),$$ where $\mathcal{O(x^3)}$ means including all the higher order ...
Stallmp's user avatar
  • 318
-1 votes
0 answers
30 views

quadrupoles etc. [closed]

I pasted maple equation input. In an asymptotic expansion, why is the last term referred to as a quadrupole. I just dont get how that can be. $F(r, \phi) = 1+2\times \frac{C_1}{r}-\frac{(C_2\times \...
diyer's user avatar
  • 1
3 votes
2 answers
182 views

On the order of growth of entire functions

Definition. Let $f: \mathbb{C} \to \mathbb{C}$ be an entire function. The order of growth of $f$, denoted by $O_G(f)$, is defined as \begin{equation} O_G(f) := \inf \left\{r > 0: \exists A, B &...
Leonidas's user avatar
  • 1,054
0 votes
0 answers
38 views

Bounded in probability and continuous function

Suppose $ X_1, X_2, … $ is a sequence of random vectors in $ \mathbb{R}^d $ that is bounded in probability. That is, for every $ \epsilon > 0$, there exists $M_\epsilon > 0$ such that: $ \mathbb{...
DensetClopen's user avatar
0 votes
0 answers
63 views

Proof using the Stirling formula for a limit behaviour

I am trying to understand/prove a lemma which is stated in two papers about branching processes (Lemma 2.2 in "Martingales And Large Deviations For Binary Search Trees" by Jabbour-Hattab and ...
CampFire's user avatar
  • 178
0 votes
0 answers
11 views

Calculating the asymptotic distribution of a function of an LLN-like quantity.

I'm preparing for my statistics exam. I'm considering the following question - there is a continuously differentiable strictly monotonic function $\mu : \mathbb{R} \rightarrow \mathbb{R}$ and a ...
Featherball's user avatar
2 votes
0 answers
28 views

Does the limit of the exponential mobius exponential series asymptotically equal its regularized power series?

Context: Consider the function $\sum_{n=0}^{\infty} e^{nx}$. An extremely unrigorous manipulation of this series would yield $$ \sum_{n=0}^{\infty} e^{nx} = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \...
Sidharth Ghoshal's user avatar
1 vote
1 answer
59 views

Is this Collatz-like stochastic process almost surely bounded?

I recently asked a question about whether a stochastic process $X =(X_n)_n$ could exhibit two properties at the same time, namely that $P(\sup_n X_n < \infty) = 1$ while $E(X_n) \to \infty$ as $n\...
Rob's user avatar
  • 6,749
0 votes
0 answers
26 views

Asymptotic Distribution of the Product of Beta and Binomial

Suppose that $X_{n} \sim \frac{Binomial(n, p)}{n}$ and $Y_{n} \sim F^{-1}(Beta(n/2 + 1/2, n/2 + 1/2))$, where $\{X\}$ and $\{Y\}$ are independent and $F: [0, 1] \rightarrow \mathbb{R}_{+}$. Define $Z_{...
Eric Weine's user avatar
5 votes
0 answers
105 views

Asymptotic estimate of the inverse Fourier transform of $e^{-\xi^{2q}}$

In V. Yu. Krylov's paper [1], he estimates the inverse Fourier transform of $e^{-\xi^{2q}}$ using harmonic measure arguments (which I don't understand since the reference is written in Russian ...
Mango Warrior's user avatar
0 votes
0 answers
12 views

Asymptotic Distribution and Describe Sources of Increasing Power in an hypothesis testing problem

I am currently dealing with the following problem in a past exam (with no solution): Suppose $S$ follows the Poisson distribution with mean $2\lambda>0$, here $\lambda$ is a parameter. Another two ...
INvisibLE's user avatar
  • 188
5 votes
2 answers
99 views

Asymptotic properties of integral of power of sine function?

I'm trying to investigate the asymptotic property of the following integral: $$ \int_0^{\theta} \sin^nx dx, \quad n\to +\infty $$ where $\theta \in (0,\pi/2)$ is a constant. For the case $\theta = ...
Romeo Liu's user avatar
1 vote
1 answer
47 views

Are these properties contradictory for a stochastic process?

Let $(X_n)_n$ be a stochastic process. Is it possible to have both that $P(\sup_n X_n < \infty) = 1$ and $E(X_n) \to \infty$ at the same time? Here is the example I have in mind: let $X_0 = 0$ and ...
Rob's user avatar
  • 6,749
0 votes
0 answers
23 views

A question on how to choose parameters such that a function decays to $-\infty$ when $n$ increases

Question how to choose $p,\delta$ such that $$n\log(\frac{100}{\delta})-\frac{n^2\delta^2p}{64(1-p)+6\delta}$$ increases when $n$ grows. Motivation I am trying to understand Corollary 5.7 in this ...
chloe's user avatar
  • 554

1
2 3 4 5
191