On convergence to non normal laws for ergodic random fields

Dalibor Voln{ý}

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arXiv · 2605.31085

On convergence to non normal laws for ergodic random fields

Dalibor Voln{ý}

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Abstract

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.

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Dalibor Voln{ý}. 2026-05-29. On convergence to non normal laws for ergodic random fields. https://arxiv.org/abs/2605.31085

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