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# Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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### Estimating the bound of the integral over whole $\mathbb{R}$ of the Taylor remainder term?

Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function which has a smooth inverse and satisfies the estimate \begin{equation} \lvert f(x) \rvert \leq \lvert x \rvert. \end{equation} Also, let $d\mu$ ...
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### You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to ...
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### How fast does this summation grow?

$n,i\in\mathbb N$. The summation in question is $$\sum_{k=1}^n\prod_{l=1}^k\binom{2^n}{2^l}^i.$$ How fast does this grow? I am specifically looking at $i=1,2$.
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### Hypothesis to guarantee Lindeberg's condition

Imagine to have a set of random variables $\{ X_i \}_{i=1}^{n}$ independent (Non identically distributed). In these scenario, if the Lindeberg's condition hold we can extend the result of the CLT, i.e....
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### How do I solve the following definite integral (preferably by an asymptotic method)?

$$\int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx$$ Note: $\mu$ here is an extremely small constant. I have tried: Estimating the integral by ...
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### Does the surface area of the unit Lp ball go to zero for all $p < \infty$?

We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...
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### Is there a "convolution" of asymptotic growth?

Suppose that I have two asymptotic counts given by $$\#\{x \in [0,H] \cap \mathbb Z: f(x) \leq H\} \sim F(H)$$ and also $$\#\{x \in [0,H] \cap \mathbb Z: g(x) \leq H\} \sim G(H).$$ From these two ... 140 views

### Precise asymptotics for moments of order statistics of normal distribution

Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...
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### applying the watson lemma to an integral [closed]

So i thought about applying the Watson lemma to determine the asymptotic behavior of the integral $$I(x)=\int_{0}^{\infty} \frac{e^{-x(t-\ln(t))}}{(1+t^2)} dt,$$ as $x \rightarrow \infty$. I think ...
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### Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?

The question is as in the title: Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...