CSE599s Counting and Sampling: Homework 1

D(t) = max x,y∈Ω ‖K(x, ·)−K(y, ·)‖TV ≤ 2 max x∈Ω ‖K(x, ·)− π‖TV = 2∆(t). Let X0 = x and Y0 = y. By the coupling lemma, there exists a coupling of K (x, ·) and K(y, ·) so that Pr[Xt 6= Yt] = ‖K(x, ·)−K(y, ·)‖TV = Dxy(t). Then for s ≥ 1, we construct a coupling of Xt+s and Yt+s as follows: • If Xt = Yt, then Xt+i = Yt+i for i = 1, ..., s. • If Xt = x′ and Yt = y′ where x′ 6= y′, then by the coupling lemma we are able to construct Xt+s and Yt+s so that Pr[Xt+s 6= Yt+s ∣∣Xt = x′, Yt = y′] = ‖Ks(x′, ·)−Ks(y′, ·)‖TV = Dx′y′(s) ≤ D(s).