# Convergence of the Population Dynamics algorithm in the Wasserstein metric

## TL;DR

This paper proves the convergence of the population dynamics algorithm in the Wasserstein metric for a class of stochastic fixed-point equations, ensuring the algorithm's reliability in approximating the special endogenous solution.

## Contribution

It establishes the convergence in Wasserstein metric and the consistency of estimators for the population dynamics algorithm applied to stochastic fixed-point equations.

## Key findings

- Convergence in Wasserstein metric of order p ($p \\geq 1$) is proven.
- Sample-based estimators are shown to be consistent.
- The results validate the algorithm's effectiveness in approximating the solution.

## Abstract

We study the convergence of the population dynamics algorithm, which produces sample pools of random variables having a distribution that closely approximates that of the {\em special endogenous solution} to a stochastic fixed-point equation of the form: $$R\stackrel{\mathcal D}{=} \Phi( Q, N, \{ C_i \}, \{R_i\}),$$ where $(Q, N, \{C_i\})$ is a real-valued random vector with $N \in \mathbb{N}$, and $\{R_i\}_{i \in \mathbb{N}}$ is a sequence of i.i.d. copies of $R$, independent of $(Q, N, \{C_i\})$; the symbol $\stackrel{\mathcal{D}}{=}$ denotes equality in distribution. Specifically, we show its convergence in the Wasserstein metric of order $p$ ($p \geq 1$) and prove the consistency of estimators based on the sample pool produced by the algorithm.