Poissonian pair correlations for dependent random variables

TL;DR

The paper extends the theory of Poissonian pair correlations (PPC) to dependent random sequences by analyzing two common dependency structures: random walks on the torus and jittered sampling. For random walks, PPC holds generically under mild smoothness (Lebesgue density) assumptions on the step distribution, with precise thresholds at $p>1$ and a convergence in expectation at $p=1$. For jittered sampling, PPC is obtained for $M$-batch extensions but not for sequential extensions, which instead exhibit weak PPC, clarifying how extension methods influence local correlation statistics. The results broaden PPC applicability to dependent settings and illuminate the role of density/yield conditions and batching in achieving almost-sure Poissonian behavior.

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.

Figures (1)

• Figure 1: The density of a normal distribution $\mathcal{N}(0.5, 0.5)$ (yellow), the multiple parts of its density mod $1$ (red) and the sum of these components, i.e. the normal distribution mod $1$ (blue).

Theorems & Definitions (17)

• Theorem 1.1
• Example 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• proof : Proof of Theorem \ref{['thm:rand_walk_ppc']}