8,674 research outputs found
Null flows, positive flows and the structure of stationary symmetric stable processes
This paper elucidates the connection between stationary symmetric
alpha-stable processes with 0<alpha<2 and nonsingular flows on measure spaces
by describing a new and unique decomposition of stationary stable processes
into those corresponding to positive flows and those corresponding to null
flows. We show that a necessary and sufficient for a stationary stable process
to be ergodic is that its positive component vanishes
Ergodic decompositions of stationary max-stable processes in terms of their spectral functions
We revisit conservative/dissipative and positive/null decompositions of
stationary max-stable processes. Originally, both decompositions were defined
in an abstract way based on the underlying non-singular flow representation. We
provide simple criteria which allow to tell whether a given spectral function
belongs to the conservative/dissipative or positive/null part of the de Haan
spectral representation. Specifically, we prove that a spectral function is
null-recurrent iff it converges to in the Ces\`{a}ro sense. For processes
with locally bounded sample paths we show that a spectral function is
dissipative iff it converges to . Surprisingly, for such processes a
spectral function is integrable a.s. iff it converges to a.s. Based on
these results, we provide new criteria for ergodicity, mixing, and existence of
a mixed moving maximum representation of a stationary max-stable process in
terms of its spectral functions. In particular, we study a decomposition of
max-stable processes which characterizes the mixing property.Comment: 21 pages, no figure
The ergodic decomposition of asymptotically mean stationary random sources
It is demonstrated how to represent asymptotically mean stationary (AMS)
random sources with values in standard spaces as mixtures of ergodic AMS
sources. This an extension of the well known decomposition of stationary
sources which has facilitated the generalization of prominent source coding
theorems to arbitrary, not necessarily ergodic, stationary sources. Asymptotic
mean stationarity generalizes the definition of stationarity and covers a much
larger variety of real-world examples of random sources of practical interest.
It is sketched how to obtain source coding and related theorems for arbitrary,
not necessarily ergodic, AMS sources, based on the presented ergodic
decomposition.Comment: Submitted to IEEE Transactions on Information Theory, Apr. 200
Indicator fractional stable motions
Using the framework of random walks in random scenery, Cohen and
Samorodnitsky (2006) introduced a family of symmetric -stable motions
called local time fractional stable motions. When , these processes
are precisely fractional Brownian motions with . Motivated by random
walks in alternating scenery, we find a "complementary" family of symmetric
-stable motions which we call indicator fractional stable motions.
These processes are complementary to local time fractional stable motions in
that when , one gets fractional Brownian motions with .Comment: 11 pages, final version as accepted in Electronic Communications in
Probabilit
Decomposability for stable processes
We characterize all possible independent symmetric alpha-stable (SaS)
components of an SaS process, 0<alpha<2. In particular, we focus on stationary
SaS processes and their independent stationary SaS components. We also develop
a parallel characterization theory for max-stable processes.Comment: Major revision. Section 4 of previous version removed due to a
mistake in the proof. Remarks 3.2 and 3.3 adde
Nonparametric inference for fractional diffusion
A non-parametric diffusion model with an additive fractional Brownian motion
noise is considered in this work. The drift is a non-parametric function that
will be estimated by two methods. On one hand, we propose a locally linear
estimator based on the local approximation of the drift by a linear function.
On the other hand, a Nadaraya-Watson kernel type estimator is studied. In both
cases, some non-asymptotic results are proposed by means of deviation
probability bound. The consistency property of the estimators are obtained
under a one sided dissipative Lipschitz condition on the drift that insures the
ergodic property for the stochastic differential equation. Our estimators are
first constructed under continuous observations. The drift function is then
estimated with discrete time observations that is of the most importance for
practical applications.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ509 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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