 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2010/m/2/c/i/solution

The chain alternates between the classes

{1, 2}

and

{3, 4}

. It is irreducible and has period two, since no odd return is possible and each state has a positive two-step return probability. Matrix multiplication gives

P^{2} = 5/9 4/9 00 4/9 5/9 00 00 5/9 4/9 00 4/9 5/9 .

The even-time sampled chain stays in its initial class and moves between its two states with probability

4/9

at each sampled step.

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