Convergence 0 1 0 1/3 P= 1/3 0 2/3 /3 1/3 1/3 1/3 0.2500 0.3750 0.3750 P20 ≈ 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 ergodic:convergent to stationary distribution
Convergence P = ⇤ 010 1/302/3 1/3 1/3 1/3 ⇥ ⌅ 1/3 1/3 1/3 1/3 2/3 1 1 2 3 P20 ⇤ 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 ⇥ ⌅ ergodic: convergent to stationary distribution
12 1-2 2-3 13 1/2 1/2 0 0 reducible 1/3 2/3 0 0 0 3/4 1/4 4 0 0 1/4 3/4 34 34 14 0.4 0.6 0 0 p20 0.4 0.6 0 0 ≈ 0 0 0.5 0.5 0 0 0.5 0.5
P = ⇧ ⇧ ⇤ 1/2 1/20 0 1/3 2/30 0 0 03/4 1/4 0 01/4 3/4 ⇥ ⌃ ⌃ ⌅ P20 ⇧ ⇧ ⇤ 0.4 0.60 0 0.4 0.60 0 0 00.5 0.5 0 00.5 0.5 ⇥ ⌃ ⌃ ⌅ 1 2 1 2 1 3 2 3 3 4 1 4 3 4 1 4 reducible
pexiodc 1 m9 p26- 69 P d
1 1 P = 0 1 1 0⇥ P2 = 1 0 0 1⇥ P2k = 1 0 0 1⇥ periodic P2k+1 = 0 1 1 0⇥
1-2 12 23 1-3 reducible 3 34 14 34 1/3 3 2/3 peiodc 1
1 / 3 1 / 3 1 / 3 1 / 3 2 / 3 1 1 2 3 12 1213 23 34 14 34 14 reducible 11 periodic
Fundamental Theorem of Markov Chain: If a finite Markov chain =(P)is irreducible and aperiodic,then initial distribution(0) (ergodic) limπ(o)pt=π t→∞ where is a unique stationary distribution satisfying πP=π
If a finite Markov chain is irreducible and aperiodic, then ∀ initial distribution M = (⌦, P) ⇡(0) lim t!1 ⇡(0)Pt = ⇡ where is a ⇡ unique stationary distribution satisfying ⇡P = ⇡ Fundamental Theorem of Markov Chain: (ergodic)