Material for “ Weak Convergence Rates of Population versus Single - Chain Stochas - Tic Approximation Mcmc Algorithms ”

Proof. Let M = sup θ∈Θ max{{h(θ), |v(θ)|} and V ε = {θ : d(θ, L) ≤ ε}. Applying Taylor's expansion formula (Folland, 1990), we have v(θ t+1) = v(θ t) + γ n+1 v h (θ t+1) + R t+1 , t ≥ 0, which implies that t i=0 γ i+1 v h (θ i) = v(θ t+1) − v(θ 0) − t i=0 R i+1 ≥ −2M − t i=0 R i+1. Since t i=0 R i+1 converges (owing to Lemma A.2), t i=0 γ i+1 v h (θ i) also converges. Furthermore, v(θ t) = v(θ 0) + t−1 i=0 γ i+1 v h (θ i) + t−1 i=0 R i+1 , t ≥ 0, {v(θ t)} t≥0 also converges. On the other hand, conditions (A 1) and (A 2) imply lim t→∞ d(θ t , L) = 0. Otherwise, there exists ε > 0 and n 0 such that d(θ t , L) ≥