Heavy tails and the failure of the central limit theorem

Heavy-tailed distributions have become very important in fields as diverse as telecommunications and economics. They often occur in situations where one would expect that the Central Limit Theorem should apply. This article investigates why the Central Limit Theorem fails, and shows one mechanism by which heavy-tailed behavior can arise. This is by addition of what the article defines as “hypercorrelated” random variables. The article also shows that heavy-tailed behavior cannot arise due to addition of linearly related random variables. The failure of the Central Limit Theorem to be applicable in many areas where it has traditionally been assumed to apply has important real-world consequences, especially in finance and financial modeling.