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A heavy-tailed distribution is often defined as a distribution with a tail that is not exponentially bounded.

A truncated power law (or power law with exponential cut-off) is a distribution that consist of a power law distribution multiplied by an exponential distribution:

$$ f(x) \propto x^{\alpha}e^{\beta x} $$

This distribution behaves alike a power law for small values of x, but experiments a heavier decrease in the probability of large values of x than a power law. Some papers describes truncated power laws behaviors as follows:

[The truncated power law] has the power law’s scaling behavior over some range but is truncated by an exponentially bounded tail.

Is the truncated power law a heavy-tailed distribution due to its power law-like behavior over the smaller x range?

Alternatively, is the exponentially bounded tail of the truncated power law enough to reject it is a heavy-tailed distribution?

http://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0085777&type=printable

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A main reason that heavy-tailed distributions are of interest is that they can model processes that produce occasional extreme values. Infinite tails are not necessary for this purpose; many (considering our measurement systems that give us actual data) real processes are both bounded and outlier-prone. Truncated distributions with truncation points sufficiently far out in tails satisfy a definition of heavy-tailedness that matches the actual intent of their use.

Classic definitions of heavy-tailedness that require infinite tails are too narrow: They do not admit a large class of very relevant distributions that are very useful for modelling data-generating processes that produce occasional extreme values; namely, bounded distributions.

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