distribution at timewhen initial state isx Markov Chain Coupling Lemma: (X,Y)is a coupling of =(2,P) △(t)≤max Pr[Xt≠Y|Xo=x,Yo=y x,y∈2 △()=竖p-xy ≤max x,y∈2 llsv ≤max Pr[Xt卡Yt|Xo=x,Yo=y c,y∈2 (coupling lemma)(Xt, Yt) is a coupling of M = (⌦, P) ￾(t)  max x,y2⌦ Pr[Xt 6= Yt | X0 = x, Y0 = y] Markov Chain Coupling Lemma: p(t) x : distribution at time t when initial state is x ￾(t) = max x2⌦ kp(t) x ￾ ⇡kT V  max x,y2⌦ kp(t) x ￾ p(t) y kT V  max x,y2⌦ Pr[Xt 6= Yt | X0 = x, Y0 = y] (coupling lemma)