11,245 research outputs found
Correlated fractional counting processes on a finite time interval
We present some correlated fractional counting processes on a finite time
interval. This will be done by considering a slight generalization of the
processes in Borges et al. (2012). The main case concerns a class of space-time
fractional Poisson processes and, when the correlation parameter is equal to
zero, the univariate distributions coincide with the ones of the space-time
fractional Poisson process in Orsingher and Polito (2012). On the other hand,
when we consider the time fractional Poisson process, the multivariate finite
dimensional distributions are different from the ones presented for the renewal
process in Politi et al. (2011). Another case concerns a class of fractional
negative binomial processes
On characterising the variability properties of X-ray light curves from active galaxies
We review some practical aspects of measuring the amplitude of variability in
`red noise' light curves typical of those from Active Galactic Nuclei (AGN).
The quantities commonly used to estimate the variability amplitude in AGN light
curves, such as the fractional rms variability amplitude, F_var, and excess
variance, sigma_XS^2, are examined. Their statistical properties, relationship
to the power spectrum, and uses for investigating the nature of the variability
processes are discussed. We demonstrate that sigma_XS^2 (or similarly F_var)
shows large changes from one part of the light curve to the next, even when the
variability is produced by a stationary process. This limits the usefulness of
these estimators for quantifying differences in variability amplitude between
different sources or from epoch to epoch in one source. Some examples of the
expected scatter in the variance are tabulated for various typical power
spectral shapes, based on Monte Carlo simulations. The excess variance can be
useful for comparing the variability amplitudes of light curves in different
energy bands from the same observation. Monte Carlo simulations are used to
derive a description of the uncertainty in the amplitude expected between
different energy bands (due to measurement errors). Finally, these estimators
are used to demonstrate some variability properties of the bright Seyfert 1
galaxy Markarian 766. The source is found to show a strong, linear correlation
between rms amplitude and flux, and to show significant spectral variability.Comment: 14 pages. 12 figures. Accepted for publication in MNRA
On the fractional Poisson process and the discretized stable subordinator
The fractional Poisson process and the Wright process (as discretization of
the stable subordinator) along with their diffusion limits play eminent roles
in theory and simulation of fractional diffusion processes. Here we have
analyzed these two processes, concretely the corresponding counting number and
Erlang processes, the latter being the processes inverse to the former.
Furthermore we have obtained the diffusion limits of all these processes by
well-scaled refinement of waiting times and jumpsComment: 30 pages, 4 figures. A preliminary version of this paper was an
invited talk given by R. Gorenflo at the Conference ICMS2011, held at the
International Centre of Mathematical Sciences, Pala-Kerala (India) 3-5
January 2011, devoted to Prof Mathai on the occasion of his 75 birthda
From infinite urn schemes to decompositions of self-similar Gaussian processes
We investigate a special case of infinite urn schemes first considered by
Karlin (1967), especially its occupancy and odd-occupancy processes. We first
propose a natural randomization of these two processes and their
decompositions. We then establish functional central limit theorems, showing
that each randomized process and its components converge jointly to a
decomposition of certain self-similar Gaussian process. In particular, the
randomized occupancy process and its components converge jointly to the
decomposition of a time-changed Brownian motion , and the randomized odd-occupancy process and its components
converge jointly to a decomposition of fractional Brownian motion with Hurst
index . The decomposition in the latter case is a special case of
the decompositions of bi-fractional Brownian motions recently investigated by
Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed
as correlated random walks, and in particular as a complement to the model
recently introduced by Hammond and Sheffield (2013) as discrete analogues of
fractional Brownian motions.Comment: 25 page
Generalized (m,k)-Zipf law for fractional Brownian motion-like time series with or without effect of an additional linear trend
We have translated fractional Brownian motion (FBM) signals into a text based
on two ''letters'', as if the signal fluctuations correspond to a constant
stepsize random walk. We have applied the Zipf method to extract the
exponent relating the word frequency and its rank on a log-log plot. We have
studied the variation of the Zipf exponent(s) giving the relationship between
the frequency of occurrence of words of length made of such two letters:
is varying as a power law in terms of . We have also searched how
the exponent of the Zipf law is influenced by a linear trend and the
resulting effect of its slope. We can distinguish finite size effects, and
results depending whether the starting FBM is persistent or not, i.e. depending
on the FBM Hurst exponent . It seems then numerically proven that the Zipf
exponent of a persistent signal is more influenced by the trend than that of an
antipersistent signal. It appears that the conjectured law
only holds near . We have also introduced considerations based on the
notion of a {\it time dependent Zipf law} along the signal.Comment: 24 pages, 12 figures; to appear in Int. J. Modern Phys
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