a)

A Markov chain is called a regular chain if some power of the transition matrix has only positive elements.

\(P= \begin{pmatrix} .5 & .5 \\ .5 & .5 \end{pmatrix}\)

Here, we’ve only got postie elements: so this is regular.

b)

\(P= \begin{pmatrix} .5 & .5 \\ 1 & 0 \end{pmatrix}\)

Here, we’ve got only positive entries with at least one diagonal not equal to zero so this is regular, too.

c)

\(P= \begin{pmatrix} 1/3 & 0 & 2/3\\ 0 & 1 & 0 \\ 0 & 1/5 & 4/5\\ \end{pmatrix}\)

For this one, I wanted to check the powers of the matrix to see if they converged:

P <- matrix(c(1/3,0,2/3,0,1,0,0,1/5,4/5), nrow=3, ncol=3, byrow = T)
P

##           [,1] [,2]      [,3]
## [1,] 0.3333333  0.0 0.6666667
## [2,] 0.0000000  1.0 0.0000000
## [3,] 0.0000000  0.2 0.8000000

P %^% 100

##              [,1] [,2]         [,3]
## [1,] 1.940325e-48    1 2.910051e-10
## [2,] 0.000000e+00    1 0.000000e+00
## [3,] 0.000000e+00    1 2.037036e-10

As shown, multiplying by itself repeatedly results in 0’s. Therefore this is not regular.