4,791 research outputs found
Universal Coding and Prediction on Martin-L\"of Random Points
We perform an effectivization of classical results concerning universal
coding and prediction for stationary ergodic processes over an arbitrary finite
alphabet. That is, we lift the well-known almost sure statements to statements
about Martin-L\"of random sequences. Most of this work is quite mechanical but,
by the way, we complete a result of Ryabko from 2008 by showing that each
universal probability measure in the sense of universal coding induces a
universal predictor in the prequential sense. Surprisingly, the effectivization
of this implication holds true provided the universal measure does not ascribe
too low conditional probabilities to individual symbols. As an example, we show
that the Prediction by Partial Matching (PPM) measure satisfies this
requirement. In the almost sure setting, the requirement is superfluous.Comment: 12 page
The dimension of ergodic random sequences
Let \mu be a computable ergodic shift-invariant measure over the Cantor
space. Providing a constructive proof of Shannon-McMillan-Breiman theorem,
V'yugin proved that if a sequence x is Martin-L\"of random w.r.t. \mu then the
strong effective dimension Dim(x) of x equals the entropy of \mu. Whether its
effective dimension dim(x) also equals the entropy was left as an problem
question. In this paper we settle this problem, providing a positive answer. A
key step in the proof consists in extending recent results on Birkhoff's
ergodic theorem for Martin-L\"of random sequences
Anomalous scaling due to correlations: Limit theorems and self-similar processes
We derive theorems which outline explicit mechanisms by which anomalous
scaling for the probability density function of the sum of many correlated
random variables asymptotically prevails. The results characterize general
anomalous scaling forms, justify their universal character, and specify
universality domains in the spaces of joint probability density functions of
the summand variables. These density functions are assumed to be invariant
under arbitrary permutations of their arguments. Examples from the theory of
critical phenomena are discussed. The novel notion of stability implied by the
limit theorems also allows us to define sequences of random variables whose sum
satisfies anomalous scaling for any finite number of summands. If regarded as
developing in time, the stochastic processes described by these variables are
non-Markovian generalizations of Gaussian processes with uncorrelated
increments, and provide, e.g., explicit realizations of a recently proposed
model of index evolution in finance.Comment: Through text revision. 15 pages, 3 figure
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