Limit law for some modified ergodic sums

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.