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 special endogenous solution to a stochastic fixed-point equation of the form: R D = Φ(Q,N, {Ci}, {Ri}), where (Q,N, {Ci}) is a real-valued random vector with N ∈ N, and {Ri}i∈N is a sequence of i.i.d. copies of R, independent of (Q,N, {Ci}); the symbol D = denotes equality in distribution. Specifically, we show its convergence in the Wasserstein metric of order p (p ≥ 1) and prove the consistency of estimators based on the sample pool produced by the algorithm.