Poissonian pair correlations for dependent random variables

TL;DR

This paper investigates Poissonian pair correlations in dependent random sequences, specifically jittered samples and random walks on the torus, revealing conditions under which PPC holds.

Contribution

It demonstrates how dependency structures affect PPC in jittered samples and proves generic PPC for random walks on the torus under mild assumptions.

Findings

PPC depends on how finite jittered samples extend to infinite sequences.

Random walks on the torus generically exhibit PPC under mild conditions.

The study links dependency structures with Poissonian pair correlation properties.

Abstract

We consider Poissonian pair correlations (PPC) for uniformly distributed sequences of random numbers with a dependency structure. More specifically, we treat two classes of dependent random variables which have widely been studied in the literature, namely sequences of jittered samples and random walks on the torus. We show that for the former class, the PPC property depends on how the finite sample is extended to an infinite sequence. Moreover, we prove that, under some mild assumptions, the random walk on the torus generically has PPC.