Markov Chain Conditional Expectation

[tunaaa]

Hello, in relation to Markov chains, could you please clarify the following equations:

In particular, could you please expand on why the first line is equal. Surely from

, along with the first equation, this implies that:

I just don't see why they are all equal. Please could you provide some intuition on this. Thanks

 

[SW VandeCarr]

Hello, in relation to Markov chains, could you please clarify the following equations:

In particular, could you please expand on why the first line is equal. Surely from

, along with the first equation, this implies that:

I just don't see why they are all equal. Please could you provide some intuition on this. Thanks

The Markov property is where the probability of the present state n is conditional only on the probability of the immediately preceding state n-1. There's no dependence on prior states n-i for integer i; $$1<i\leq n$$.

 

[tunaaa]

OK, sure - but how is that fact relevant to understanding the above equation? Thanks.

 

[SW VandeCarr]

OK, sure - but how is that fact relevant to understanding the above equation? Thanks.

Given the Markov property, if $$P(X_{n}=j|(X_{0}=j)$$, then under what two conditions could this be true?