Probability LOWER LIMITS AND EQUIVALENCES FOR CONVOLUTION TAILS

Put γ̂ = sup{γ : φ(γ) < ∞} ∈ [0,∞]. Note that the function φ(γ) is monotone continuous in the interval (−∞, γ̂), and φ(γ̂) = lim γ↑γ̂ φ(γ) ∈ [1,∞]. We distinguish all the distributions on [0,∞) according to the value of γ̂. If γ̂ = 0, then we say that the distribution F is heavy-tailed; in that case φ(γ) = ∞ for any γ > 0. If γ̂ > 0, then we call the distribution F light-tailed; this happens if and only if, for some γ > 0, F (x) = o(e−γx) as x →∞. The main results of this paper are the following Theorems 1, 2, and 3 which relate the tail behaviour of the convolution F ∗ F to that of F .