Nonincreasing of Total Variation p):distribution at time t when initial state isx △z(t)=lp-πlrv Theorem: △x(t+1)≤△x(t) consider y~π △(=lp-prv x=X1 X2,....Xn X+1 -Y1,Y2,...YY coupling lemma:3 a coupling (Xi,Y)s.t.Pr[X]=A(t) couple (X1,Y+1)in such a way that:X=YX=Y △x(t)=Pr[X≠Y≥Pr[X+1≠Y+]≥lp+D-πlTv =△x(t+1)Nonincreasing of Total Variation ￾x(t) = kp(t) x ￾ ⇡kT V p(t) x : distribution at time t when initial state is x Theorem: ￾x(t + 1)  ￾x(t) consider y ∼ π ￾x(t) = kp(t) x ￾ p(t) y kT V coupling lemma: ∃ a coupling (Xt, Yt) s.t. x = X1, X2, ..., Xt, Xt+1, ... π ∼ Y1, Y2, ..., Yt, Yt+1, ... Pr[Xt 6= Yt] = ￾x(t) couple (Xt+1, Yt+1) in such a way that: Xt=Yt ⇒ Xt+1=Yt+1 ￾x(t) = Pr[Xt 6= Yt] ￾ Pr[Xt+1 6= Yt+1] ￾ kp(t+1) x ￾ ⇡kT V = ￾x(t + 1)