On permutation-invariance of limit theorems

TL;DR

This paper investigates the conditions under which subsequences of random variables maintain their probabilistic and analytic properties under permutation, extending previous work on trigonometric systems to general sequences.

Contribution

It generalizes the study of permutation-invariance from trigonometric systems to arbitrary random variable sequences, linking it to Diophantine properties.

Findings

Permutation-invariance depends on Diophantine properties of index sequences.

Rearrangement can alter the analytic properties of lacunary sequences.

Extension of previous results from trigonometric systems to general random sequences.

Abstract

By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d.\ sequences. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis, such the theory of orthogonal series and Banach space theory. In contrast to i.i.d.\ sequences, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change after rearrangement. In a previous paper we showed that permutation-invariance of subsequences of the trigonometric system and related function systems is connected with Diophantine properties of the index sequence. In this paper we will study permutation-invariance of subsequences of general r.v.\ sequences.