Conditional independence and conditioned limit laws

Conditioned limit laws constitute an important and well developed framework of extreme value theory that describe a broad range of extremal dependence forms including asymptotic independence. We explore the assumption of conditional independence of $X_1$ and $X_2$ given $X_0$ and study its implication in the limiting distribution of $(X_1,X_2)$ conditionally on $X_0$ being large. We show that under random norming, conditional independence is always preserved in the conditioned limit law but might fail to do so when the normalisation does not include the precise value of the random variable in the conditioning event.