can does anybody know if the following expectations are available in closed for...
Let $\{ X_t : t = 1, 2, 3 \dots \}$ be a random variable defined on a Markov chain with m -step transition matrix $P_m^{i,j}$. I'm trying hunting for a closed form expression for the following expectations which are telescoping:
$m = 1 : E_t \Big [ \frac{1}{X_{t+1}} \Big ]$
$m = 2 : E_t \Big [ \frac{1}{X_{t+1}} \frac{1}{X_{t+2}} \Big ]$
$m = 3 : E_t \Big [ \frac{1}{X_{t+1}} \frac{1}{X_{t+2}} \frac{1}{X_{t+3}}\Big ]$
A clue would be lovely - thanks.