A value for alpha of 2 corresponds to the normal distribution, with its "nonmonstrous," well-understood, mathematically tractable properties and rapidly diminishing tails. It is a form of "mild" randomness: the rapidly diminishing tails mean that extreme events are very infrequent. If alpha is less than 2, however, the tails of the distribution are sufficiently "fat" that the statistician's standard ways of measuring the extent of a distribution's "spread" (the standard deviation and its square, the variance) no longer exist: the integral expression that defines them does not converge, and so they have no finite value and are in that sense infinite.

Donald MacKenzie, An Engine, Not a Camera, loc. 480