Irreducible and recurrent Markov chain - theorem notation question

Question

In [J. R. Norris] Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics) (2009), page 35, Theorem 1.7.5 says:

In (ii), does it mean $\gamma^k$ is notation for $\gamma^k_i$ over all $i \in I$ or for some fixed arbitrary $i \in I$?

I think it's the latter but I just want to make sure that I'm interpreting this nice and compact way of writing it correctly.

Answer 1

Your first interpretation is correct: for example if $I=\{1,2,3\}$, then $\gamma^k$ is short for $(\gamma_1^k,\gamma_2^k,\gamma_3^k)$. Another way this is notated is $(\gamma_i^k)_{i \in I}$.