1,235 research outputs found
Central limit theorems and suppression of anomalous diffusion for systems with symmetry
We give general conditions for the central limit theorem and weak convergence
to Brownian motion (the weak invariance principle / functional central limit
theorem) to hold for observables of compact group extensions of nonuniformly
expanding maps. In particular, our results include situations where the central
limit theorem would fail, and anomalous behaviour would prevail, if the compact
group were not present.
This has important consequences for systems with noncompact Euclidean
symmetry and provides the rigorous proof for a conjecture made in our paper: A
Huygens principle for diffusion and anomalous diffusion in spatially extended
systems. Proc. Natl. Acad. Sci. USA 110 (2013) 8411-8416.Comment: Minor revision
Broadband nature of power spectra for intermittent Maps with summable and nonsummable decay of correlations
We present results on the broadband nature of the power spectrum ,
, for a large class of nonuniformly expanding maps with
summable and nonsummable decay of correlations. In particular, we consider a
class of intermittent maps with
for , where . Such maps have summable decay of
correlations when , and extends to a
continuous function on by the classical Wiener-Khintchine Theorem.
We show that is typically bounded away from zero for H\"older
observables.
Moreover, in the nonsummable case , we show that
is defined almost everywhere with a continuous extension defined on , and is typically
nonvanishing.Comment: Final versio
On the detection of superdiffusive behaviour in time series
We present a new method for detecting superdiffusive behaviour and for
determining rates of superdiffusion in time series data. Our method applies
equally to stochastic and deterministic time series data (with no prior
knowledge required of the nature of the data) and relies on one realisation (ie
one sample path) of the process. Linear drift effects are automatically removed
without any preprocessing. We show numerical results for time series
constructed from i.i.d. -stable random variables and from deterministic
weakly chaotic maps. We compare our method with the standard method of
estimating the growth rate of the mean-square displacement as well as the
-variation method, maximum likelihood, quantile matching and linear
regression of the empirical characteristic function
- β¦