First Passage Decomposition

[stukbv]

Homework Statement

I have a Stochastic Processes test coming up soon and we are taught about the first passage decomposition. I understand this but it then says that it changes it to a generating function

Homework Equations

Generating function P_ij(s) = Ʃ p_ij⁽ⁿ⁾sⁿ
and F_ij(s) = Ʃ f_ij⁽ⁿ⁾sⁿ


What does this generating function actually mean and what is the difference between
P_ij(s) and F_ij(s)?

 

[Ray Vickson]

Homework Statement

I have a Stochastic Processes test coming up soon and we are taught about the first passage decomposition. I understand this but it then says that it changes it to a generating function

Homework Equations

Generating function P_ij(s) = Ʃ p_ij⁽ⁿ⁾sⁿ
and F_ij(s) = Ʃ f_ij⁽ⁿ⁾sⁿ


What does this generating function actually mean and what is the difference between
P_ij(s) and F_ij(s)?

Well, p_ij⁽ⁿ⁾ and f_ij⁽ⁿ⁾ are different things, so of course their generating functions are different. The generating function does not have to mean anything; it is just a useful tool in the arsenal of problem-solving resources. However in the case of F_ij(s) we can assign a meaning, for what it is worth: F_ij(s) = E[s^T_ij], where T_ij = the first passage time from state i to state j, at least in the case where such passage is certain (i.e., Pr{T_ij < ∞} = 1.)

RGV