# Is the truncated power law a heavy-tailed distribution?

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