Dalibor Voln{ý} | Searcharxiv

arXiv2026

On convergence to non normal laws for ergodic random fields

For Z d actions equipped with a completely commuting filtration, normalized partial sums of martingale differences converge in distribution (see [V19]) but even if the action is ergodic, the limit law need not be normal. Explication of this phenomenon has been studied in [GLV] but often in dimension 2 only. It turned out that the non normality is related to the factor of product type I. Here we extend the necessary and sufficient condition of normality from dimension 2 to all finite dimensions, as well as a sufficient condition of normality for martingale differences in the orthocomplement of L 2 (I). At the same time an improved and corrected version of the central limit theorem for random fields from [V19] with dimension d > 2 is given.

math.PR↗

arXiv2022

On the central limit theorem for stationary random fields under L 1 -projective condition

The first aim of this paper is to wonder to what extent we can generalize the central limit theorem of Gordin [5] under the so-called L 1-projective criteria to ergodic stationary random fields when completely commuting filtrations are considered. Surprisingly it appears that this result cannot be extended to its full generality and that an additional condition is needed.

math.PR↗