Characteristic Functions and the Tail Behavior of Probability Distributions
The theory of Fourier transforms of generalized functions is used to extract general formulas for the tail behavior of a probability distribution from the behavior of its characteristic function in the locality of the origin. The theory is applied to develop asymptotic formula for the tails of the stable distributions.
Phillips, Peter C.B., "Characteristic Functions and the Tail Behavior of Probability Distributions" (1980). Cowles Foundation Discussion Papers. 803.