Questions tagged [slowly-varying-functions]
For questions about the so-called slowly varying functions, usually defined on the positive real line and whose "variation" is small (in particular, they grow slower than any power). This class contains, for example, all the powers of logarithms.
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Prove $V$ is slowly-varying if $\frac{V(tx)-V(t)}{a(t)} \to \log x$ (excercise 0.4.3.1 in Extreme Values, Regular Variation and Point Processes)
Given $V \colon (0,\infty) \to (0,\infty)$ monotone increasing, such that for all $x>0$, $\lim_{t \to \infty} \frac{V(tx) - V(t)}{a(t)} = \log x$ for some function $a(t) > 0$, how can we prove ...
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Reguarly varying function and convergence in probability
The following is an exercise from Extreme Values, Regular Variation and Point Processes.
We say a function $f \colon (0,\infty) \to (0,\infty)$ is regularly varying with index $\rho$ if for all $x>...
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Slowlying varying function with no limit at infinity: using Karamata representation theorem
I want to find a slowly varying function that has no limit at infinity, and write down its Karamata's representation .
That is, I want to find $L: (0,\infty) \to (0,\infty)$ with the $\lim_{x \to \...
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If $\lim_{n \to \infty} \lambda_n U(a_n x) \to \chi(x)$, $U$ is regulary varying and $\frac{\chi(x)}{\chi(1)} = x^\rho$
The following is a problem from Extreme Values, Regular Variation and Point Processes by Resnick.
We say $U \colon (0,\infty) \to (0,\infty)$ is regularly-varying if for all $x>0$, $\lim_{t \to \...
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Prove $\exp(\int^x_0 \frac{U(t)}{t} dt)$ is reguarly varying (Extreme Values, Regular Variation and Point Processes)
I want to prove that if $\lim_{x \to \infty} \frac{1}{x} \int^x_0 U(t) dt = \rho \in \mathbb{R}$, then $\exp(\int^x_0 \frac{U(t)}{t} dt)$ is regularly varying with $\rho$.
Looking in Seneta, it is ...
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Prove for monotone $L \colon (0, \infty) \to (0,\infty)$, $L$ is slowly-varying if and only if there exists $x$ such that $\frac{L(tx)}{L(t)} \to 1$
$L \colon (0,\infty) \to (0,\infty)$ is slowly varying if for all $x>0$, $x \neq 1$ $ \lim_{t \to \infty} \frac{L(tx)}{L(t)} = 1$.
I want to prove that for if $L$ is monotone, $L$ is slowly varying ...
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Why is $L(x) = x^{o(1)}$ for any slowly varying function $L$?
Question
I've heard that for any slowly varying function $L$, it's true that $L(x) = x^{o(1)}$. But why is this the case?
Note: Here I'm using the standard (Karamata) definition of slow variation, ...
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Inequality with slowly varying functions
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Let $X$ be a random variable with distribution function $F$ on a probability space $(\Omega, \mathcal F, P)$.
Suppose that there exist $\alpha \in (0,2)$ and a slowly varying function $\ell(\...
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Which of the following conditions should be weaker?
Let $g:[0, \infty) \rightarrow \mathbb{R}$ and $h:[0, \infty) \rightarrow \mathbb{R}$ be nonnegative functions let us put $\phi(y)=g(y) h(y)$. Assume that the following condition is satisfied:
$h$ is ...
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How to prove the property of slowly varying function ''$\displaystyle\lim_{x\to +\infty}x^{-\delta}L(x)=0$ for any $\delta>0$"?
As we kown, a positive function $L: [0,+\infty)\to\mathbb{R}^+$ is called slowly varying function if
$$\lim_{x\to+\infty}\frac{L(cx)}{L(x)}=1 \text{ for any $c>0$}. $$
I find the property of slowly ...
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Is the ''proof" for a proposition about slowly varying function correct?
The positive function $f: [0,+\infty)\to\mathbb{R}^+$ is called slowly varying function if
$$\lim_{x\to+\infty}\frac{f(cx)}{f(x)}=1 \text{ for any $c>0$}. $$
Now there is a ''proof" for this ...
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Time Average of a function $f(\phi(t),t)$, $\phi(t)$ is only varying very slowly with time.
I am a bit stumped on a task. Namely I have the function:
\begin{align}
f(\phi(t),t) = \cos{(\omega (t-t_0))} \, \cos{\phi} + \cos^2{(\omega (t-t_0))}\,\sin{\phi}\,.
\end{align}
And my task is to ...
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Is there a function which grows faster than any slowly varying function but slower than any power of x?
I was thinking about if there are random variables lying in between different domains of attractions. And in short, it seems to me, that this latter question boils down to: is there a function which ...
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Slowly-varying functions near zero
It is well known that, if $x\mapsto f(x)$ is a slowly-varying function, then
$$
\lim_{x\to\infty}\sup_{\lambda\in K}\left|\frac{f(\lambda x)}{f(x)}-1\right| = 0,
$$
where $K\subset (0,\infty)$ is a ...
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A basic property of slowly varying functions
It seems that a fundamental property of a slowly varying function is that for all $\delta > 0$
$$ \lim_{x\to \infty} L(x) \, x^{-\delta} = 0.$$
How to prove this? The only book I found (that was ...
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A question related to slowly oscillating functions from De-Koninck&Luca-Analytic NT
Let $f:\mathbb{N}\to\mathbb{C}$ be a function for which there exists a constant $0<A\in\mathbb{R}$ such that: $\lim_{x\to\infty}\frac{1}{x}\sum_{n\le x}f(n)=A.$
Show that:
$\lim_{x\to\infty}\sum_{n\...
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Question related to asymptotic equivalence and possibly regularly varying functions
Let $f(n)$, $n \in \mathbb{N}$, be a function such that $f(n) \sim n^{-1/2}$. Let $\delta \in (0,1)$. I want to prove that $$ \sum_{j \leq \lfloor \delta n \rfloor} f(j) \sim 2 \delta n \, f(\lfloor \...
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Examples of Continuous Probability Distributions with Slowly Varying Upper Tail and Infinite Expectation [closed]
$L(x) = 1 - F(x)$ is the slowly varying upper tail, where F(x) is a continuous probability distribution function with infinite expectation. That is $\int_{0}^{\infty} u dF(u) = \infty$. I am unable to ...
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Are Quasi-Polynomial Time Functions Regularly Varying?
Is $$\lim_{x\to\infty} \frac{e^{(\ln ax)^k}}{e^{(\ln x)^k}}$$ finite for all positive real $a$ and $k$?
I have tried this on Desmos with $a$ and $k$ less than 2 where it seems to converge after ...
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Slowly varying functions and the direction of convergence
Consider a slowly varying function $L(n)$ and let $\lambda > 0$. Is it true that there exists a $N \in \mathbb{N}$ such that
\begin{align}
\frac{L(\lambda n)}{L(n)} \geq 1
\end{align}
for all $n \...
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Is a nondecreasing function bounded by a slowly varying function also slowly varying?
I have a seemingly simple question that I haven't been able to resolve:
Question. Suppose that $f:(0,\infty)\to(0,\infty)$ is nondecreasing. Suppose that $f\leq L$ for a slowly varying function $L$....
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regular and slowly varying functions examples $e^{\log(x)}$, $e^{\lfloor \log(x) \rfloor}$, $2 + \sin(x)$ and $e^{(\log(x))^\beta}$
A function $f$ is called regular varying with level $\alpha$ if for any $\lambda > 0$ it holds
$\lim_{x \rightarrow \infty} \frac{f(\lambda x)}{f(x)} = \lambda ^\alpha$.
A regular varying ...
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For which values of $\alpha \geq 0$ is the function $\exp(\log^\alpha(x))$ varying regularly?
I am trying to prove for which values of $\alpha \geq 0$ the following function $f(x) := \exp(\log^\alpha(x))$ is varying regularly (or slowly) and if it is varying, I need to determine the index of ...
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On uniform convergence of bounded slowly varying functions
A positive function $f: \mathbb{R} \to \mathbb{R}$ is said to be slowly varying if$$\lim \limits_{t \to \infty} \frac{f(tx)}{f(t)}=1$$
for all $x >0$. Assume that $f(x) \in [m,M]$ for all $x$. Is ...
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Composite of slowly varying functions
A function, $L$ is slowly varying if for all $b>0$:
$$
\lim_{x \to \infty} \frac{L(bx)}{L(x)}=1.
$$
If two functions, $L_1, L_2$ are slowly varying, and we have further that $L_2(x) \to \infty$ ...
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Slowly varying function without limit at infinity
A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$
$$
\lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1.
$$
Is there a bounded function slowly varying at infinity whose ...
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