On permutations of lacunary series

TL;DR

This paper investigates how permuting terms in lacunary series affects their asymptotic properties, revealing conditions under which classical limit theorems like CLT and LIL remain valid or change.

Contribution

It provides necessary and sufficient criteria for permutation-invariance of the CLT and LIL in lacunary series, clarifying when asymptotic behaviors are preserved under rearrangements.

Findings

Permutation-invariance of CLT and LIL depends on specific criteria.

Rearrangements can alter the asymptotic behavior of lacunary series.

The paper characterizes when classical limit theorems hold after permutation.

Abstract

It is a well known fact that for periodic measurable $f$ and rapidly increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a sequence of independent, identically distributed random variables. For example, if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies the central limit theorem, the law of the iterated logarithm and several further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence remains i.i.d.\ after any permutation of its terms, it is natural to expect that the asymptotic properties of lacunary series are also permutation-invariant. Recently, however, Fukuyama (2009) showed that a rearrangement of the sequence $(f(2^kx))_{k\ge 1}$ can change substantially its asymptotic behavior, a very surprising result. The purpose of the present paper is to investigate this interesting phenomenon in detail and to give necessary and sufficient criteria for the permutation-invariance of the CLT and LIL for $f(n_kx)$.