# Universality for conditional measures of the Bessel point process

## TL;DR

This paper proves that the conditional measures of the Bessel point process converge to the original process as the interval expands to infinity, demonstrating a form of universality in the process's behavior.

## Contribution

It establishes the almost sure convergence of conditional measures of the Bessel point process to the original process as the interval size grows, for a broad class of measures.

## Key findings

- Conditional measures are orthogonal polynomial ensembles.
- Convergence occurs almost surely as R tends to infinity.
- The result applies to a deterministic class of probability measures.

## Abstract

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval $[0,R]$ is almost surely an orthogonal polynomial ensemble. In this article, we show that if $R$ tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure of the Bessel point process almost surely belongs.