Limit law for some modified ergodic sums
Jean-Pierre Conze (IRMAR), St\'ephane Le Borgne (IRMAR)

TL;DR
This paper generalizes a classical example showing that normalized sums of certain functions along lacunary sequences can have mixed Gaussian distributions, by analyzing modified ergodic sums within dynamical systems.
Contribution
It introduces a new framework for understanding the limiting behavior of modified ergodic sums, extending classical results to broader dynamical systems.
Findings
Normalized sums can have mixed Gaussian limit laws
Generalization of Erdos-Fortet example to dynamical systems
Provides insight into distributional limits of lacunary sequences
Abstract
An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems.
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