Exercise: Periodic states
2/4 points (graded)
Consider a Markov chain with the following transition probability graph:
This figure depicts a Markov chain with eleven states. The possible states, which are depicted as circles with the state number inside them, are 1, 2, 3, up to 11. In this figure, transitions are depicted by arrows from a circle (representing the source) to a circle (representing the destination); the source and the destination may be the same. The eleven states are arranged in three rows. States 1, 2, 3, 4, 5, and 6 form a three-by-two rectangular block on the left side of the figure, with 1 and 2 on the first row left and right, 3 and 4 on the second row left and right, 5 and 6 on the third row left and right. On the first row, there is an arrow from 1 to 2; on the second row, there is an arrow from 3 to 4; and on the third row, there are arrows both from 5 to 6 and from 6 to 5. In between the first two rows are arrows from 1 to 3 and from 3 to 1, as well as arrows from 2 to 4 and from 4 to 2. In between the second and third rows are arrows from 5 to 3 and from 4 to 6. In the middle column of the figure, at the second and third rows are states 7 and 8. There is an arrow from state 7 to 8 and from state 8 to 7. Connecting this middle column and the left threeby-two block are arrows from 7 to 4 (on the second row) and from 8 to 6 (on the third row). At the right of the figure there are is a rhombus consisting of four states: 9, 10, 11, and 12. State 10 is on the top row, 9 and 12 on the middle row (left and right), and 11 at the bottom row. To help visualize the location of the circles, if we consider the projection onto the x axis, it is 9, then 10 and 11, then 12. There is an arrow from 10 to 9, from 11 to 9, from 12 to 10, and from 12 to 11, as well as an arrow from 9 to 12. Connecting this rhombus to the middle column, there are arrows from 7 to 9 (within the second row), from 8 to 11 (within the third row), and from 8 to 9 (stretching from the third to the secondrow diagonally up).
How many recurrent classes are there?
2
correct
How many periodic recurrent classes are there?
1
incorrect
What is the smallest period among these?
2
correct
What is the largest period among these?