1,323 research outputs found
Randomness on computable probability spaces - A dynamical point of view
We extend the notion of randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff’s pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications
Randomness on Computable Probability Spaces - A Dynamical Point of View
We extend the notion of randomness (in the version introduced by Schnorr) to
computable Probability Spaces and compare it to a dynamical notion of
randomness: typicality. Roughly, a point is typical for some dynamic, if it
follows the statistical behavior of the system (Birkhoff's pointwise ergodic
theorem). We prove that a point is Schnorr random if and only if it is typical
for every mixing computable dynamics. To prove the result we develop some tools
for the theory of computable probability spaces (for example, morphisms) that
are expected to have other applications
Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
A pseudorandom point in an ergodic dynamical system over a computable metric
space is a point which is computable but its dynamics has the same statistical
behavior as a typical point of the system.
It was proved in [Avigad et al. 2010, Local stability of ergodic averages]
that in a system whose dynamics is computable the ergodic averages of
computable observables converge effectively. We give an alternative, simpler
proof of this result.
This implies that if also the invariant measure is computable then the
pseudorandom points are a set which is dense (hence nonempty) on the support of
the invariant measure
Compression and diffusion: a joint approach to detect complexity
The adoption of the Kolmogorov-Sinai (KS) entropy is becoming a popular
research tool among physicists, especially when applied to a dynamical system
fitting the conditions of validity of the Pesin theorem. The study of time
series that are a manifestation of system dynamics whose rules are either
unknown or too complex for a mathematical treatment, is still a challenge since
the KS entropy is not computable, in general, in that case. Here we present a
plan of action based on the joint action of two procedures, both related to the
KS entropy, but compatible with computer implementation through fast and
efficient programs. The former procedure, called Compression Algorithm
Sensitive To Regularity (CASToRe), establishes the amount of order by the
numerical evaluation of algorithmic compressibility. The latter, called Complex
Analysis of Sequences via Scaling AND Randomness Assessment (CASSANDRA),
establishes the complexity degree through the numerical evaluation of the
strength of an anomalous effect. This is the departure, of the diffusion
process generated by the observed fluctuations, from ordinary Brownian motion.
The CASSANDRA algorithm shares with CASToRe a connection with the Kolmogorov
complexity. This makes both algorithms especially suitable to study the
transition from dynamics to thermodynamics, and the case of non-stationary time
series as well. The benefit of the joint action of these two methods is proven
by the analysis of artificial sequences with the same main properties as the
real time series to which the joint use of these two methods will be applied in
future research work.Comment: 27 pages, 9 figure
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