2
$\begingroup$

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.

  • 0
    ok, or even more generally, how would one find : $E_t [\prod_{i=0}^k f(x_{t+i})] $2012-10-11
  • 0
    The problem formulation is not clear. If $X$ is a Markov Chain? Does it take values over reals (otherwise, what is $1/X$)? What is $E_t$?2012-12-04

1 Answers 1