Say the [KL divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) between two distributions $A$ and $B$ is $\varepsilon$. Can we give bounds, or a precise computation, of the KL divergence between $A^k$ and $B^k$ (the product distributions)?



Kullback–Leibler divergence of product distributions
Transforming usul's comment into a proper answer: if the KL divergence between $A$ and $B$ is $\varepsilon$, the KL divergence between $A^k$ and $B^k$ is $k\varepsilon$. This follows directly from the chain rule (Theorem 5.3 of [this PDF](http://www.math.tau.ac.il/~mansour/advanced-agt+ml/scribe5-lower-bound-MAB.pdf), applied to a situation where $x$ and $y$ are independent).


