Conditional independence and conditioned limit laws

TL;DR

This paper investigates how the assumption of conditional independence affects the limiting distribution of variables in extreme value theory, highlighting the importance of normalization choices in preserving this independence.

Contribution

It demonstrates that conditional independence is preserved under random norming but may not hold with fixed normalization, clarifying implications for conditioned limit laws.

Findings

Conditional independence is preserved under random norming.

Failure of conditional independence can occur with fixed normalization.

Implications for modeling extremal dependence in multivariate extremes.

Abstract

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.