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

2023/12/24 by Dmitry Dolgopyat, Dolgopyat, Dmitry, Sixu Liu +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2312.15378

openalex publication_date 2023/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)^α, 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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