# Long-time limits and occupation times for stable Fleming-Viot processes with decaying sampling rates

## TL;DR

This paper introduces and analyzes a class of Fleming-Viot processes with decaying sampling rates and stable motions, establishing long-time behavior, occupation times, and asymptotics for growing populations, with broader applicability to measure-valued processes.

## Contribution

It develops almost sure long-time scaling limits and asymptotics for Fleming-Viot processes with decaying sampling rates and stable motions, extending understanding of population distribution over time.

## Key findings

- Established almost sure long-time scaling limits.
- Analyzed occupation and inhabitation time convergence.
- Extended techniques to general Feller motion/mutation processes.

## Abstract

A class of Fleming-Viot processes with decaying sampling rates and $\alpha$-stable motions that correspond to distributions with growing populations are introduced and analyzed. Almost sure long-time scaling limits for these processes are developed, addressing the question of long-time population distribution for growing populations. Asymptotics in higher orders are investigated. Convergence of particle location occupation and inhabitation time processes are also addressed and related by way of the historical process. The basic results and techniques allow general Feller motion/mutation and may apply to other measure-valued Markov processes.