Abstract: We present a unifying approach for comparing and classifying tails of
heavy-tailed random variables. The main result establishes, for any
heavy-tailed random variable, existence of a certain concave function
representing the asymptotic decay speed of the tail. Many key
properties of the distribution of a random variable are encoded into
this function. Our approach extends the idea of classical indices,
such as exponential and moment indices, which are widely used
measuring heaviness of tails.