314 research outputs found
Convergence to stable laws in the space
We study the convergence of centered and normalized sums of i.i.d. random
elements of the space of c{{\'a}}dl{{\'a}}g functions endowed
with Skorohod's topology, to stable distributions in . Our
results are based on the concept of regular variation on metric spaces and on
point process convergence. We provide some applications, in particular to the
empirical process of the renewal-reward process
Estimating the scaling function of multifractal measures and multifractal random walks using ratios
In this paper, we prove central limit theorems for bias reduced estimators of
the structure function of several multifractal processes, namely mutiplicative
cascades, multifractal random measures, multifractal random walk and
multifractal fractional random walk as defined by Lude\~{n}a [Ann. Appl.
Probab. 18 (2008) 1138-1163]. Previous estimators of the structure functions
considered in the literature were severely biased with a logarithmic rate of
convergence, whereas the estimators considered here have a polynomial rate of
convergence.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ489 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Function-indexed empirical processes based on an infinite source Poisson transmission stream
We study the asymptotic behavior of empirical processes generated by
measurable bounded functions of an infinite source Poisson transmission process
when the session length have infinite variance. In spite of the boundedness of
the function, the normalized fluctuations of such an empirical process converge
to a non-Gaussian stable process. This phenomenon can be viewed as caused by
the long-range dependence in the transmission process. Completing previous
results on the empirical mean of similar types of processes, our results on
non-linear bounded functions exhibit the influence of the limit transmission
rate distribution at high session lengths on the asymptotic behavior of the
empirical process. As an illustration, we apply the main result to estimation
of the distribution function of the steady state value of the transmission
process
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