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