# Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices

## TL;DR

This paper completely solves the ergodic decomposition problem for inverse Wishart measures on infinite positive-definite matrices, extending the classification of invariant measures in infinite-dimensional matrix spaces.

## Contribution

It provides the first complete ergodic decomposition for inverse Wishart measures, a significant step in understanding invariant measures on infinite matrices.

## Key findings

- Complete ergodic decomposition for inverse Wishart measures achieved
- Extends classification of invariant measures to a new family
- Provides foundational results for infinite-dimensional matrix analysis

## Abstract

The ergodic unitarily invariant measures on the space of infinite Hermitian matrices have been classified by Pickrell and Olshanski-Vershik. The much-studied complex inverse Wishart measures form a projective family, thus giving rise to a unitarily invariant measure on infinite positive-definite matrices. In this paper we completely solve the corresponding problem of ergodic decomposition for this measure.