We recently went through expected hitting times of markov chains in my class and were asked about computing the various moments of hitting times.
As such, I'm wondering if my thinking is correct as far as computing the third moment of a hitting time.
With transition matrix $P$, we denote the third moment as such:
$E_y[T^3] = E[(T-1)^3 | X_1 = y]$
$= \sum_{y \in c} P(X_1 = y | X_0 = x)(E_y[T^3] - 3E_y[T^2] + 3E_y[T] + 1)$
where $E_y[T^2]$ and $E_y[T]$ are the second and first moments which would already have to have been computed in order to compute the third(call them $v, u)$
this lets us set up a system of equations such that the second and first moments are vectors that let us solve for the third (call it $m$)
$= \sum_{y \in c} P(X_1 = y | X_0 = x)(m(y) - 3v(y) + 3u(y) + 1)$
$= 1 + \sum_{y \in c} P(X_1 = y | X_0 = x)(m(y) - 3v(y) + 3u(y))$
and thus we can find the solution by solving:
$m(y) = (I-P)^{-1}(1- 3v(y) + 3u(y))$
Is my thinking correct? Any additional intuition would be most appreciated.
Thank you