89 research outputs found
Probabilistic sampling of finite renewal processes
Consider a finite renewal process in the sense that interrenewal times are
positive i.i.d. variables and the total number of renewals is a random
variable, independent of interrenewal times. A finite point process can be
obtained by probabilistic sampling of the finite renewal process, where each
renewal is sampled with a fixed probability and independently of other
renewals. The problem addressed in this work concerns statistical inference of
the original distributions of the total number of renewals and interrenewal
times from a sample of i.i.d. finite point processes obtained by sampling
finite renewal processes. This problem is motivated by traffic measurements in
the Internet in order to characterize flows of packets (which can be seen as
finite renewal processes) and where the use of packet sampling is becoming
prevalent due to increasing link speeds and limited storage and processing
capacities.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ321 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Exponents, symmetry groups and classification of operator fractional Brownian motions
Operator fractional Brownian motions (OFBMs) are zero mean, operator
self-similar (o.s.s.), Gaussian processes with stationary increments. They
generalize univariate fractional Brownian motions to the multivariate context.
It is well-known that the so-called symmetry group of an o.s.s. process is
conjugate to subgroups of the orthogonal group. Moreover, by a celebrated
result of Hudson and Mason, the set of all exponents of an operator
self-similar process can be related to the tangent space of its symmetry group.
In this paper, we revisit and study both the symmetry groups and exponent
sets for the class of OFBMs based on their spectral domain integral
representations. A general description of the symmetry groups of OFBMs in terms
of subsets of centralizers of the spectral domain parameters is provided. OFBMs
with symmetry groups of maximal and minimal types are studied in any dimension.
In particular, it is shown that OFBMs have minimal symmetry groups (as thus,
unique exponents) in general, in the topological sense. Finer classification
results of OFBMs, based on the explicit construction of their symmetry groups,
are given in the lower dimensions 2 and 3. It is also shown that the
parametrization of spectral domain integral representations are, in a suitable
sense, not affected by the multiplicity of exponents, whereas the same is not
true for time domain integral representations
Wavelet-based simulation of fractional Brownian motion revisited
We reexamine the wavelet-based simulation procedure for fractional Brownian motion proposed by Abry and Sellan. We clarify in what sense the wavelet-based simulation procedure works, shed light on the structure of associated fractional low- and high-pass filters, and consequently suggest some modifications to the simulation algorithm
Nonminimal sets, their projections and integral representations of stable processes
New criteria are provided for determining whether an integral representation of a stable process is minimal. These criteria are based on various nonminimal sets and their projections, and have several advantages over and shed light on already available criteria. In particular, they naturally lead from a nonminimal representation to the one which is minimal. Several known examples are considered to illustrate the main results. The general approach is also adapted to show that the so-called mixed moving averages have a minimal integral representation of the mixed moving average type
Nonminimal sets, their projections and integral representations of stable processes
New criteria are provided for determining whether an integral representation of a stable process is minimal. These criteria are based on various nonminimal sets and their projections, and have several advantages over and shed light on already available criteria. In particular, they naturally lead from a nonminimal representation to the one which is minimal. Several known examples are considered to illustrate the main results. The general approach is also adapted to show that the so-called mixed moving averages have a minimal integral representation of the mixed moving average type
Stable stationary processes related to cyclic flows
We study stationary stable processes related to periodic and cyclic flows in
the sense of Rosinski [Ann. Probab. 23 (1995) 1163-1187]. These processes are
not ergodic. We provide their canonical representations, consider examples and
show how to identify them among general stationary stable processes.
We conclude with the unique decomposition in distribution of stationary
stable processes into the sum of four major independent components: 1.
A mixed moving average component. 2. A harmonizable (or ``trivial'')
component.
3. A cyclic component 4. A component which is different from these.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000010
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