Infinite Determinantal Measures and The Ergodic Decomposition of Infinite Pickrell Measures II. Convergence of determinantal measures

TL;DR

This paper investigates the convergence behavior of infinite determinantal measures, providing conditions for tightness and convergence of sequences, with implications for understanding infinite Pickrell measures and their ergodic decompositions.

Contribution

It introduces new criteria for the convergence of infinite determinantal measures, expanding the theoretical framework for analyzing infinite point processes.

Findings

Established sufficient conditions for tightness of determinantal measure families

Proved convergence criteria for sequences of induced processes

Analyzed convergence under finite-dimensional perturbations

Abstract

The second part of the paper mainly deals with convergence of infinite determinantal measures, understood as the convergence of the approximating finite determinantal measures. In addition to the usual weak topology on the space of probability measures on the space of configurations, we also consider the weak topology on the space of finite measures on the space of finite measures on the half-line, used via the natural immersion, well-defined almost surely with respect to the infinite Bessel point process, of the space of configurations into the space of finite measures on the half-line. The main results of the second part are sufficient conditions for tightness of families of determinantal measures, for convergence of sequences of induced proceses, as well as for sequences of finite-dimensional perturbations of determinantal processes.