Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Hurst roughness exponent and its model-free estimation   We say that a continuous real-valued function $x$ admits the Hurst roughness
exponent $H$ if the $p^{\text{th}}$ variation of $x$ converges to zero if
$p>1/H$ and to infinity if $p<1/H$. For the sample paths of many stochastic
processes, such as fractional Brownian motion, the Hurst roughness exponent
exists and equals the standard Hurst parameter. In our main result, we provide
a mild condition on the Faber--Schauder coefficients of $x$ under which the
Hurst roughness exponent exists and is given as the limit of the classical
Gladyshev estimates $\widehat H_n(x)$. This result can be viewed as a strong
consistency result for the Gladyshev estimators in an entirely model-free
setting, because no assumption whatsoever is made on the possible dynamics of
the function $x$. Nonetheless, our proof is probabilistic and relies on a
martingale that is hidden in the Faber--Schauder expansion of $x$. Since the
Gladyshev estimators are not scale-invariant, we construct several
scale-invariant estimators that are derived from the sequence $(\widehat
H_n)_{n\in\mathbb N}$. We also discuss how a dynamic change in the Hurst
roughness parameter of a time series can be detected. Finally, we extend our
results to the case in which the $p^{\text{th}}$ variation of $x$ is defined
over a sequence of unequally spaced partitions. Our results are illustrated by
means of high-frequency financial time series.

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The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin