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Register a new userConvergence of the Population Dynamics algorithm in the Wasserstein metric
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Authors
Mariana Olvera-Cravioto
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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.
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We analyze the convergence properties of the Wang-Landau algorithm. This sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias. Such algorithms are very helpful to enhance the sampling properties of Markov Chain Monte Carlo algorithms, when the dynamics is metastable. We prove the convergence of the Wang-Landau algorithm and an associated central limit theorem.
The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.
We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $infty-$Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slepv{c}ev to the case that the true distribution has an unbounded density.
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