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The aim of this note is to give a straightforward proof of a general version of the Ciesielski–Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski–Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly, a new transformation which maps a subset of the family of Laplace exponents of spectrally negative Lévy processes into itself. Secondly, some classical features...
Given a two-dimensional fractional multiplicative process determined by two Hurst exponents and , we show that there is an associated uniform Hausdorff dimension result for the images of subsets of by if and only if .
We study the asymptotic behavior of the empirical process when the underlying data are gaussian and exhibit seasonal long-memory. We prove that the limiting process can be quite different from the limit obtained in the case of regular long-memory. However, in both cases, the limiting process is degenerated. We apply our results to von–Mises functionals and -Statistics.
We study the asymptotic behavior of the empirical process when the
underlying data are Gaussian and exhibit seasonal
long-memory. We prove that the limiting process can be quite
different from the limit obtained in the case of regular
long-memory. However, in both cases, the limiting process is
degenerated. We apply our results to von–Mises functionals and
U-Statistics.
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