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Questions tagged [p-variation]

Use this tag for questions about $p$-variation norms, the study of finiteness of the $p$- variation of functions, which is a generalization of the total variation.

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Topology on space of full signatures in rough path theory

In Theorem 3.1 of this paper, the following result is formulated: Suppose $f:S_1 \to \mathbb{R}$ is a continuous function where $S_1$ is a compact subset of $S(\mathcal{V}^p(J,E))$. Then for any $\...
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Two-way anova or One-way anova

I have to tested multiple reinforcement learning systems and record their performances [test-accuracy] based on two categories of [system-type] and [communication-level]. The system-type can be either ...
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How to show the 2-variation of Brownian motion sample paths is infinite

Brownian motion has bounded quadratic variation, however for almost every sample path, the $p$-variation is infinite for any $p>1/2$, where the $p$ variation takes the supremum over all possible ...
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Link between $p-variation$ and $Holder$ norm

I'm studying some basic notions of Young Integration and I got stuck with the so called "Link between $p-variation$ and $Holder$ norm" found on the wikipedia page: the criminal page. In particular ...
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Why is $\lim_{n\to\infty} 2^n \Psi\left(\frac{r}{2^n}\right)=0$, for some specific function $\Psi$ defined in the question?

This comes from the proof of the first theorem in this blog article. The paths $x:[0,T]\to\mathbb{R}^d$ and $y:[0,T]\to\mathbb{R}^{e\times d}$ are of bounded total variation. The real numbers $p,q>...
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Inequality involving $p-$variation and $\sup$ norms.

Let's call $\mathcal D(\Bbb R_{\ge0},\Bbb R^d)$ the Skohrokod space, i.e. the space of the functions $f:\Bbb R_{\ge0}\to\Bbb R^d$ continous on the right which admit limit on the left. Let's fix $y,l\...
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On the $p$-variation norm: is $||\cdot||_{p-var}\le||\cdot||_{\infty}$?

Let $\Bbb D(\Bbb R_{\ge0},\Bbb R^d)$ be the space of the Càdlàg functions. If $p\ge1$ and $x\in\Bbb D(\Bbb R_{\ge0},\Bbb R^d)$, we define the norm $$ \overline V_p(x)_T:=|x_0|+\left(\sup_{\pi}\sum_{j=...