Convergence of the Population Dynamics algorithm in the Wasserstein metric

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: $$Rstackrel{mathcal D}{=} Phi( Q, N, { C_i }, {R_i}),$$ 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.