Question

1. Let (т, P) be a time-homogeneous discrete-time Markov chain with state space {1, . . . , (a) Show that the Markov chain is not stationary (i.e., SSS). (b) Suppose P is doubly stochastic and π- JJ, . . . , Đ. Then show that the Markov chain is stationary

Answer 1

1) Solution:

Given data :

($\prod , P$) be a time - homogeneous discrete - time

Markov chain state space { 1,...., J)

Now, we have to find out the :

a) Show that the Markov chain is not stationary :

i.e., $X_{t+l} = X_{t+1}+ X_{t+2} + .....+ X_{t+t} = X_{T+1 / \lambda _{t} } + X_{t/ \alpha _{-1}- \alpha _{i}/ t}$

• Markov chain isn't stationary. As a matter of first importance we need to comprehend what is stationary .
• Morkov chain, I expected in short way a morkov anchor is said to be stationary.
• In the event that it is does not relies upon the time that mean it is measurable qualities does not change over the time.
• So previously mentioned discrete morkov chain rely upon the time.
• That implies its qualities that are co-difference mean auto-relationship , auto-correlogram and so on., are changed over the time.
• So given senes isn't stationary ( by utilization of stationary definition.)

b)Suppose P is doubly stochastic and $\prod = ( \frac{1}{J}, \frac{1}{J} , .... ,\frac{1}{ J} )$ , Then show that the markov chain is stationary :

• In second circumstance pi is two fold stochastics and $\prod = ( \frac{1}{J}, \frac{1}{J} , .... ,\frac{1}{ J} )$ afterward the morkov chain is free from the parameters time t.
• That implies here the morkov chain does not relies upon the time then its arrangement is called morkov chain.