# 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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### 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 ...
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 ...