Show that the transition matrices in (4.1.4) do not satisfy Doeblin’s condition, while the matrix in Example 4.2-4 does.

Example 4.2-4

In a country, at the end of each year, the government arranges a poll where each interviewee evaluates her/his financial status as either “good,” or “fair,” or “bad”. The actual income corresponding to these states may be different for different people, but we assume that for each citizen, the process of transition from one state to another constitutes a Markov chain with the same transition matrix