Convergence of the Population Dynamics algorithm in the Wasserstein metric

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 $p$ ($p\ge 1$) and prove the consistency of estimators based on the sample pool produced by the algorithm.