6 ideas
In fat-tailed distributions, the sample mean is a poor estimator of the true mean because a single extreme observation can dominate the entire sample, and rare large events that haven't occurred yet are not reflected in past data. The empirical average converges to the true value so slowly that for many such distributions it is practically unobservable within any realistic sample size.
Phenomena divide into two domains: Mediocristan, where no single observation meaningfully changes the aggregate (height, weight, calorie intake), and Extremistan, where one observation can dominate the total (wealth, book sales, casualties). The domain determines which statistical tools are valid — averaging and Gaussian methods work only in Mediocristan and become dangerous in Extremistan.
Under fat tails the Law of Large Numbers still operates but requires vastly more observations to deliver stable estimates, sometimes orders of magnitude more than the Gaussian case. This means standard sample sizes that appear adequate produce wildly unreliable conclusions, and confidence in statistical claims should scale with tail thickness.
Because catastrophic events are rare, the absence of observed disaster in a track record is not evidence of safety — the most dangerous distributions are precisely those that look calm for long stretches before producing a single dominating event. Track records should be read as samples that systematically underrepresent the tail that matters most.
Time averages (what one trajectory experiences over time) diverge from ensemble averages (what a population experiences at one moment) when ruin is possible, because an absorbing barrier like bankruptcy or death ends the sequence permanently. A bet with positive expected value can still guarantee eventual ruin for any individual who repeats it, making the ensemble expectation irrelevant to real decisions.
The tail exponent (alpha) of a power-law distribution dictates which moments are finite: below 2 the variance is infinite, below 1 even the mean is undefined. Before applying any statistical method one must establish the tail exponent, because tools assuming finite variance produce meaningless results when those moments don't exist.