1000 times and find approximately the distribution after 10

1000 times and find approximately the distribution after 10 steps. Is it close to the stationary

distribution [1/19.5, 5/19.5, 13.5/19.5] ?

Solution

> T=rbind(c(0.5,0.3,0.2),c(0.1,0.4,0.5),c(0,0.2,0.8));T
[,1] [,2] [,3]
[1,] 0.5 0.3 0.2
[2,] 0.1 0.4 0.5
[3,] 0.0 0.2 0.8
> U=cbind(1/3,1/3,1/3);U
[,1] [,2] [,3]
[1,] 0.3333333 0.3333333 0.3333333
> t1=U%*%T;t1
[,1] [,2] [,3]
[1,] 0.2 0.3 0.5
> t2=t1%*%T;t2
[,1] [,2] [,3]
[1,] 0.13 0.28 0.59
> t3=t2%*%T;t3
[,1] [,2] [,3]
[1,] 0.093 0.269 0.638
> t4=t3%*%T;t4
[,1] [,2] [,3]
[1,] 0.0734 0.2631 0.6635
> t5=t4%*%T;t5
[,1] [,2] [,3]
[1,] 0.06301 0.25996 0.67703
> t6=t5%*%T;t6
[,1] [,2] [,3]
[1,] 0.057501 0.258293 0.684206
> t7=t6%*%T;t7
[,1] [,2] [,3]
[1,] 0.0545798 0.2574087 0.6880115
> t8=t7%*%T;t8
[,1] [,2] [,3]
[1,] 0.05303077 0.2569397 0.6900295
> t9=t8%*%T;t9
[,1] [,2] [,3]
[1,] 0.05220936 0.256691 0.6910996
> t10=t9%*%T;t10
[,1] [,2] [,3]
[1,] 0.05177378 0.2565591 0.6916671
> p=cbind(1/19.5, 5/19.5, 13.5/19.5);p
[,1] [,2] [,3]
[1,] 0.05128205 0.2564103 0.6923077

t11=t10%*%T;t11
[,1] [,2] [,3]
[1,] 0.0515428 0.2564892 0.691968
> t12=t11%*%T;t12
[,1] [,2] [,3]
[1,] 0.05142032 0.2564521 0.6921276
> t13=t12%*%T;t13
[,1] [,2] [,3]
[1,] 0.05135537 0.2564325 0.6922122
> t14=t13%*%T;t14
[,1] [,2] [,3]
[1,] 0.05132093 0.256422 0.692257