this relates to an unanswered question I posted a few days ago:
Let $\{ X_t : t = 1, 2, 3 \dots \}$ follow a 2-state Markov chain with transition matrix P. Does the Markov property mean I can break following expectations up as follows:
$E_t \Big [ X_{t+1}\ X_{t+2}\Big ] = E_t [ X_{t+1}]\ E_t[ X_{t+2}]$
$E_t \Big [ X_{t+1}\ X_{t+2}\ X_{t+3} \Big ] = E_t [ X_{t+1}]\ E_t[X_{t+2}]\ E_t[X_{t+3} ]$
Intuitively the answer is 'no' since I now have path dependent expectations?
Thanks, Paul