Functions of a Markov Chains

Question

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.

Answer 1

By definition, $$ E\Big [ \frac{1}{X_{t+1}} \frac{1}{X_{t+2}} \Big ]=\sum_{x,y,z}P[X_t=x]\frac{P_1^{x,y}P_1^{y,z}}{yz}, $$ where $$ P[X_t=x]=\sum_{u}P[X_0=u]P_t^{u,x}. $$