While the error on a probability can be small and controlled, the error on the corresponding quantity can be infinite.

This is a mathematical claim with a 'can be' (existence) structure. TRUE: standard probability theory shows a case where the estimation error on a probability, such as an exceedance probability, is bounded or small, while the error on the corresponding quantity, such as an expected payoff, partial moment or loss, is unbounded or infinite. Examples include fat-tailed distributions with tail exponent at or below 1 or 2, where the mean or variance is infinite. MOSTLY TRUE: the result holds, but only under fat-tail conditions that the claim implies rather than states, or 'infinite' should strictly read 'unbounded' or 'arbitrarily large'. MOSTLY FALSE: the corresponding-quantity error is large but always finite under standard assumptions, so 'infinite' overstates it. FALSE: the error on the quantity is always controlled whenever the error on the probability is.

Set before the research began.

1. E1 Numerical Treatment of Heavy-Tailed Distributions riskhub.org · 23 September 2026

   “Since 2 > 1.5, the variance does not exist. Not "is large". Does not exist. The defining integral diverges.”

2. E2 Taylor's law of fluctuation scaling for semivariances and higher moments of heavy-tailed data PNAS · 12 November 2021

   “When the tail index is small enough, the mean and all higher moments of the random quantity are infinite.”

3. E3 Numerical Treatment of Heavy-Tailed Distributions riskhub.org · 23 September 2026

   “Two exceedances out of 100,000 is not an estimate of a 3.16 × 10−5 probability; it is a coin-flip that landed near the right order of magnitude.”

4. E4 Numerical Treatment of Heavy-Tailed Distributions riskhub.org · 23 September 2026

   “Note also that these standard errors are meaningful, unlike the ones in section 5, because the weighted estimator has genuinely finite variance by construction even though the underlying variable does not.”

E1 and E2 show that variance or higher moments can be infinite under heavy tails, but no evidence item shows that the error on a corresponding probability is small or controlled in such a case: E4 concerns a weighted estimator without saying it estimates a probability, and E3 shows a naive tail-probability estimate being unreliable.