Order of Limits using Replica Trick

Question

I'm trying to use the replica trick ($Z\rightarrow Z^{n})$ to evaluate a particular partition function (over an $N\times N$ matrix), and I've hastily assumed that I can commute the limits of $N\rightarrow \infty$ and $n\rightarrow 0$, but upon further reflection I realize this may not be true. What conditions must be satisfied for these limits to be interchangeable? 

Comments

the existence of an absolute convergent majorant (cf. dominated convergence and Fubini's theorem).