# Simulating an infinite mean waiting time

## TL;DR

This paper introduces a hybrid simulation method for critical birth-death processes with infinite mean return times, combining direct simulation with tail sampling from a power-law distribution.

## Contribution

It proposes a novel hybrid approach to efficiently simulate return times in processes with infinite mean, addressing challenges in traditional methods.

## Key findings

- The return time distribution follows a power-law asymptotically.
- The method effectively captures rare long return times.
- Simulation results validate the hybrid approach's accuracy.

## Abstract

We consider a hybrid method to simulate the return time to the initial state in a critical-case birth--death process. The expected value of this return time is infinite, but its distribution asymptotically follows a power-law. Hence, the simulation approach is to directly simulate the process, unless the simulated time exceeds some threshold and if it does, draw the return time from the tail of the power law.