Heavy-tailed distribution

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In probability theory, heavy-tailed distributions are probability distributions with infinite variance.

The distribution of a random variable X is said to have a heavy tail if

\Pr[X>x] \sim x^{- \alpha}\mbox{ as }x \to \infty,\qquad 0< \alpha <2.\,

This means that regardless of the distribution for small values of the random variable, if the asymptotic shape of the distribution is hyperbolic, it is heavy-tailed. The simplest heavy-tailed distribution is the Pareto distribution which is hyperbolic over its entire range.

A characteristic of long-tailed distributions is that the log-log plot of the tail of a long-tailed distribution is approximately linear over many orders of magnitude [1]. If the logarithm of the range of an exponential distribution is found, the resulting plot is linear. In contrast, that of the heavy-tail distribution is still curvilinear.

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