415 research outputs found
Local limit theorem in deterministic systems
We show that for every ergodic and aperiodic probability preserving system,
there exists a valued, square integrable function such that
the partial sums process of the time series satisfies the lattice local limit theorem.Comment: 17 page
Relative Complexity of Random Walks in Random Scenery in the absence of a weak invariance principle for the local times
We answer the question of Aaronson about the relative complexity of Random
Walks in Random Sceneries driven by either aperiodic two dimensional random
walks, two-dimensional Simple Random walk, or by aperiodic random walks in the
domain of attraction of the Cauchy distribution. A key step is proving that the
range of the random walk satisfies the F\"olner property almost surely.Comment: 19 page
The orbital equivalence of Bernoulli actions and their Sinai factors
Given a countable amenable group G and 0 < L < 1, we give an elementary
construction of a type-III:L Bernoulli group action. In the case where G is the
integers, we show that our nonsingular Bernoulli shifts have independent and
identically distributed factors.Comment: 45 pages; minor revision
Finitary isomorphisms of Brownian motions
Ornstein and Shields (Advances in Math., 10:143-146, 1973) proved that
Brownian motion reflected on a bounded region is an infinite entropy Bernoulli
flow and thus Ornstein theory yielded the existence of a measure-preserving
isomorphism between any two such Brownian motions. For fixed h >0, we construct
by elementary methods, isomorphisms with almost surely finite coding windows
between Brownian motions reflected on the intervals [0, qh] for all positive
rationals q.Comment: Published at https://doi.org/10.1214/19-AOP1412 in the Annals of
Probability by the Institute of Mathematical Statistic
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