# Questions tagged [asymptotics]

The asymptotics tag has no usage guidance.

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### Asymptotic Behaviour of Solutions to a Riccati-type ODE with Small Forcing Term

Consider the following ODE:
$$f'(r) = f^2(r) + O \left( \frac{1}{r^4} \right)$$
as $r$ goes to infinity. The initial conditions are $f(1) = C <0$.
What is the behaviour of a solution $f$ at ...

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### $\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on the boundary and at infinity

Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta_g f = 0$ ...

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67 views

### An oscillatory integral estimate

Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...

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### Large time behavior of Girsanov type Geometric Brownian Motion with time-dependent drift and diffusion

Recall the Geometric Brownian Motion $X={\rm e}^{\mu W+\left(\sigma-\frac{\mu^2}{2}\right)t}$. If $\sigma<\frac{\mu^2}{2}$, then $X$ tends to 0 almost surely. But if we consider the following case,
...

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### Oscillatory integrals

Consider the integrals
$$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$
I would like to know the asymptotic behavior of $I_n(\zeta,\epsilon)$ for ...

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### Asymptotic value of the Shannon entropy

I would like to evaluate the asymptotic value of the following sum:
$$\frac{1}{2^N}\sum_{n=0}^{N} \binom{N}{n} \log_{2} \binom{N}{n}$$
This is related to the computation of the Shannon entropy. Any ...

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### Lower bound of q pochhammer symbol [closed]

How one could prove, that q pochhammer symbol $(1,1/n) = \prod_{k = 1}^{\infty}(1-\frac{1}{n^k}) \geq 1 - \frac{1}{n-1}$

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### Asymptotic of integral $\int_{1}^{e^n}(1-\frac{\ln(x)}{n})^n\,dx$

How could we find the large-$n$ asymptotic of $$\int_{1}^{e^n}\left(1-\frac{\ln x}{n}\right)^n\,dx.$$
I have a suspicion that this is $\sqrt{n}$.

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### Asymptotics for the sums from the inclusion-exclusion principle

What is a method to compute the asymptotics of a sum resulting from the inclusion-exclusion principle? Each term of the sum can be approximated perhaps by Stirling's formula or the Gaussian ...

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### Asymptotics of orbits on graphs

Let $X$ be a connected, locally finite graph with vertex set $V(X)$ and $G$ a group acting freely on $X$ such that $X/G$ is a finite graph. Fix a vertex $x$ and for $k\in\mathbb N$ set
$$
N(k)=\#\{ g\...

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### An asymptotic behavior of a sequence of special polynomials

For $n\to\infty$, I would like to know the asymptotic behavior of the polynomials defined in terms of the Gauss hypergeometric series:
$$
p_{n}(z):={}_{2}F_{1}(-n,-nz+\alpha;1;\beta),
$$
where $\alpha,...

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### Do kissing numbers with distance $d$ always grow polynomially or exponentially in dimension?

Let $A_d(n)$ be the largest number of points that can be packed on the $n$-unit sphere, such that every point is at least $d$ apart. Compare with, for instance, https://arxiv.org/abs/1507.03631
When ...

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### Recurrence relation for the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation
$$ \left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$
as $y\...

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### Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$.
Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$.
I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$.
My first ...

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54 views

### Order statistics of correlated bivariate Gaussian

Suppose $(X_1,Y_1),...,(X_n,Y_n)$ are i.i.d. bivariate Gaussian with mean zero. Each coordinate has variance 1 and correlation between coordinates is $\rho\in[-1,1]$.
I'm interested in the following ...