On the convergence rate of pair correlations
Abstract
Abstract In this note, we propose to study the rate at which the weak pair correlation statistic with parameter $$0 < \alpha \le 1$$ 0 < α ≤ 1 converges to its limit in the Poissonian case. As an application we show that if $$(x_n)_{n \in \mathbb {N}}$$ ( x n ) n ∈ N has weak $$\alpha $$ α -Poissonian pair correlations and the rate of convergence is locally uniform of order $$cN^{-\kappa }$$ c N - κ , then it also has weak $$\beta $$ β -Poissonian pair correlations for some $$\beta > \alpha $$ β > α depending on $$\kappa $$ κ . This may be regarded as a partial converse of the well-known fact that $$\beta $$ β -Poissonian pair correlations imply $$\alpha $$ α -Poissonian pair correlations if $$\beta > \alpha $$ β > α .
Fields of Study & Subjects
Random Matrices and Applications
Probability and Risk Models
Approximation Theory and Sequence Spaces
Publication details
- Journal / source
- Ricerche di Matematica
- Volume
- 75
- Issue
- 2
- Pages
- 749–757
- Publication date
- 4 January 2026
- Language
- EN
- Document type
- Book chapter