Unverified50% confidenceFactExact time
A regularly varying tail index t ∈ (1,2) means infinite variance but finite mean, and the normalized sum converges to a t-stable distribution rather than a normal distribution.
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50%
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Long-term
Relevance
10/19/2026
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Related Claims
UnverifiedThe smaller the degrees of freedom of the t-distribution, the heavier the tails; as the degrees of freedom approach infinity, the t-distribution converges to the standard normal distribution.80% similarUnverifiedThe Cauchy distribution is a special case of the t-distribution with degrees of freedom ν=1; its expected value and variance do not exist, the sample mean does not converge to a stable value, and the CLT fails in this case.78% similarUnverifiedIn the infinite-variance case t ∈ (1,2), the central fluctuation scale of SGD is of order η^{1/t}, and the limit is a t-stable distribution.77% similarUnverifiedHeavy-tailed distributions refer to distributions whose tail probabilities decay slower than exponential speed, such as the Pareto distribution, Student's t distribution, and log-normal distribution; higher-order moments of such distributions may not exist or may diverge, leading to infinite variance.69% similarUnverifiedExchangeability is a weaker requirement than independent and identically distributed (i.i.d.), requiring only that any permutation of the data points has the same joint probability distribution.68% similar
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