Given the KL divergence value between 2 distributions, how should someone use this to determine whether the value is significant for the distributions $P$ and $Q$ to be different? One method I can speculate is that a Monte Carlo sample of the CDF of the $P$ distribution can produce a range of KL distances in which there is a pvalue set at the border for the 5% largest values.
From the motivation and background for the G-test, wiki page, it shows the connection between KL divergence and the Chisquared test and the G-test. There is a clear pvalue for the Chi squared test, in that it can be applied for hypothesis testing. Is there an equivalent for KL divergence (or even the G-test)?
EDIT: the situation is that $P$ and $Q$ are sampled distributions and that they both contain long consistent tails.