Infinite-mean models in risk management: Discussions and recent advances

In statistical analysis, many classic results require the assumption that models have finite mean or variance, including the most standard versions of the laws of large numbers and the central limit theorems. Such an assumption may not be completely innocent, and it may fail for datasets with heavy tails (e.g., catastrophic losses), relevant to financial risk management. In this paper, we discuss the importance of infinite-mean models in economics, finance, and related fields, with recent results and examples. We emphasize that many results or intuitions that hold for finite-mean models turn out to fail or even flip for infinite-mean models. Due to the breakdown of standard thinking for infinite-mean models, we argue that if the possibility of using infinite-mean models cannot be excluded, great caution should be taken when applying classic methods that are usually designed for finite-mean cases in finance and insurance.