Question

3. Consider a Markov chain with state space 0, 1, 2, ... and transition probabilities p_{i,i-1} = 1, i = 1, 2, 3, ... p_{0,i} = p_i, i = 0, 1, 2, 3, ... where p_i > 0 for all i and sum_{i ge 0} p_i = 1. (c) What is the period of state i, for all values of i? (d) Under what condition is the chain positive recurrent? (e) If the chain is positive recurrent, what is the mean number of steps required for it to return to state i if it starts from i?

          3. Consider a Markov chain with state space 0, 1, 2, ... and transition probabilities
p_{i,i-1} = 1, i = 1, 2, 3, ...
p_{0,i} = p_i, i = 0, 1, 2, 3, ...
where p_i > 0 for all i and sum_{i ge 0} p_i = 1.
(c) What is the period of state i, for all values of i?
(d) Under what condition is the chain positive recurrent?
(e) If the chain is positive recurrent, what is the mean number of steps required for it to return to state i if it starts from i?

3. Consider a Markov chain with state space 0, 1, 2, ... and transition probabilities
pi,i-1 = 1, i = 1, 2, 3, ...
p0,i = pi, i = 0, 1, 2, 3, ...
where pi > 0 for all i and sumi ge 0 pi = 1.
(c) What is the period of state i, for all values of i?
(d) Under what condition is the chain positive recurrent?
(e) If the chain is positive recurrent, what is the mean number of steps required for it to return to state i if it starts from i?

Added by Ivan B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

05/03/2023

Step 1

The period of a state is the greatest common divisor (GCD) of the lengths of all cycles that include the state. In this case, we have the following cycles: - State 0: (0, 1, 2) with length 3 - State 1: (1, 2) with length 2 - State 2: (2, 1) with length 2 So, Show more…

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Consider a Markov chain with state space 0, 1, 2, ... and transition probabilities p_{i,i-1} = 1, i = 1, 2, 3, ... p_{0,i} = p_i, i = 0, 1, 2, 3, ... where p_i > 0 for all i and sum_{i >= 0} p_i = 1. (c) What is the period of state i, for all values of i? (d) Under what condition is the chain positive recurrent? (e) If the chain is positive recurrent, what is the mean number of steps required for it to return to state i if it starts from i?