Regularly varying probability densities

The convolution of regularly varying probability densities is proved asymptotic to their sum, and hence is also regularly varying. Extensions to rapid variation, O-regular variation, and other types of asymptotic decay are also given. Regularly varying distribution functions have long been used in probability theory; see e.g. Feller [7, VIII.8], Bingham, Goldie and Teugels [5, Ch. 8]. This note addresses some questions on regularly varying probability densities that seem— surprisingly—to have been overlooked. 1. Convolution of regularly varying densities Theorem 1.1. If f and g are probability densities on R, both regularly varying at ∞, then their convolution has the property (1.1) f ∗ g(x) f(x) + g(x) → 1 (x → ∞), and so is regularly varying (with index the maximum of the index of f and the index of g).