 Product large-deviation principle

Suppose $X_N$ and $Y_N$ are independent random variables in Polish spaces, each satisfying a large deviation principle at speed $a_N$, with good rate functions $I$ and $K$. Their pair has good rate function $I(x)+K(y)$. On open rectangles, independence makes the logarithmic lower bounds add. For the upper bound, the marginal families are exponentially tight, and so is their pair. Cover a closed set intersected with the product of the two compact containment sets by finitely many small rectangles and add the corresponding local upper bounds. The finite union bound and then increasing the containment level give the closed-set bound. Goodness follows because $\{I+K\le M\}$ is closed inside $\{I\le M\}\times \{K\le M\}$.