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arXiv · 2312.15378

An analogue of Law of Iterated Logarithm for Heavy Tailed Random Variables

Dmitry Dolgopyat·Sixu Liu
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Abstract

We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^\alpha$, the partial sum process has limit points consisting precisely of increasing piecewise constant functions with finitely many jumps. Our approach combines trimming techniques with a multiple Borel-Cantelli argument. It provides a functional law of the iterated logarithm for heavy-tailed processes where classical almost sure invariance principles do not apply.

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Dmitry Dolgopyat, Sixu Liu. 2023-12-24. An analogue of Law of Iterated Logarithm for Heavy Tailed Random Variables. https://arxiv.org/abs/2312.15378

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