I'm refering to the question: Differences of consecutive hitting times. I'm interested in the independence of consequtive hitting times of certain values of a Markov chain. And I do "understand" the argument in the answer, as it seems totally reasonable that this sequence is id. Nevertheless I cannot figure out a proper formal proof....
Main problem: Wy does it suffice to show that two such differences $\tau_k-\tau_{k-1}$ and $\tau_{l}-\tau_{l-1}$ for $k\neq l$ are independent? I.e. why is pairwise independence in this special case equivalent to mutual independence?
I think I can figure out a formal proof for two increments. But I'm not sure (outline: using the strong Markov property and assuming that wlog $l>k$, we know that $\tau_l-\tau_{l-1}$ is independent of $\mathcal{F}_{\tau_{l-1}}$, while $\tau_{k}-\tau_{k-1}$ is $\mathcal{F}_{\tau_{l-1}}\subset\mathcal{F}_{\tau_{k}}$-mb. Hence they are independent.)
A few nice inputs/ideas would totally suffice, I don't need a rigorous proof!
Thank's in advice!