5,254 research outputs found
Ergodic properties of Poissonian ID processes
We show that a stationary IDp process (i.e., an infinitely divisible
stationary process without Gaussian part) can be written as the independent sum
of four stationary IDp processes, each of them belonging to a different class
characterized by its L\'{e}vy measure. The ergodic properties of each class
are, respectively, nonergodicity, weak mixing, mixing of all order and
Bernoullicity. To obtain these results, we use the representation of an IDp
process as an integral with respect to a Poisson measure, which, more
generally, has led us to study basic ergodic properties of these objects.Comment: Published at http://dx.doi.org/10.1214/009117906000000692 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Disjointness properties for Cartesian products of weakly mixing systems
For we consider the class JP() of dynamical systems whose every
ergodic joining with a Cartesian product of weakly mixing automorphisms
() can be represented as the independent extension of a joining of the
system with only coordinate factors. For we show that, whenever
the maximal spectral type of a weakly mixing automorphism is singular with
respect to the convolution of any continuous measures, i.e. has the
so-called convolution singularity property of order , then belongs to
JP(). To provide examples of such automorphisms, we exploit spectral
simplicity on symmetric Fock spaces. This also allows us to show that for any
the class JP() is essentially larger than JP(). Moreover, we
show that all members of JP() are disjoint from ergodic automorphisms
generated by infinitely divisible stationary processes.Comment: 24 pages, corrected versio
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