Could anyone give me some practical examples of the Cauchy Distribution? What makes it so popular?

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In addition to its usefulness in physics, the Cauchy distribution is commonly used in models in finance to represent deviations in returns from the predictive model. The reason for this is that practitioners in finance are wary of using models that have light-tailed distributions (e.g., the normal distribution) on their returns, and they generally prefer to go the other way and use a distribution with very heavy tails (e.g., the Cauchy). The history of finance is littered with catastrophic predictions based on models that did not have heavy enough tails in their distributions. The Cauchy distribution has sufficiently heavy tails that its moments do not exist, and so it is an ideal candidate to give an error term with extremely heavy tails.

Note that this issue of the fatness of tails in error terms in finance models was one of the main contents of the popular critique by Taleb (2007). In that book, Taleb points out instances where financial models have used the normal distribution for error terms, and he notes that this underestimates the true probability of extreme events, which are particularly important in finance. (In my view this book gives an exaggerated critique, since models using heavy-tailed deviations are in fact quite common in finance. In any case, the popularity of this book shows the importance of the issue.)