# example of a case where "full conditionals" do not give a
# consistant limit dist:

# states
00  0
01  1
10  2
11  3

P1_matrix(scan(),ncol=4,byrow=T)
  .4  0  .6  0
   0 .2   0 .8
  .4  0  .6  0
   0 .2   0 .8

P2_matrix(scan(),ncol=4,byrow=T)
  .3 .7  0  0
  .3 .7  0  0
   0  0 .2 .8
   0  0 .2 .8

matpower_ function(P, pow) {
        if(pow == 1) return(P)
        else return(P %*% matpower(P, pow - 1))
}

P12_P1%*%P2
P21_P2%*%P1
options(expressions=1000)
matpower(P12,30)
[1,] 0.07346939 0.1714286 0.1510204 0.6040816
[2,] 0.07346939 0.1714286 0.1510204 0.6040816
[3,] 0.07346939 0.1714286 0.1510204 0.6040816
[4,] 0.07346939 0.1714286 0.1510204 0.6040816

matpower(P21,30)
[1,] 0.08979592 0.155102 0.1346939 0.6204082
[2,] 0.08979592 0.155102 0.1346939 0.6204082
[3,] 0.08979592 0.155102 0.1346939 0.6204082
[4,] 0.08979592 0.155102 0.1346939 0.6204082