Subexponential Densities of Infinitely Divisible Distributions on the Half-Line
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Lithuanian Mathematical Journal
Subexponential densities of infinitely divisible distributions on the half-line Toshiro Watanabe Center for Mathematical Sciences, The University of Aizu, Ikkimachi Tsuruga, Aizu-Wakamatsu, Fukushima 965-8580, Japan (e-mail: [email protected]) Received September 3, 2019; revised January 31, 2020
Abstract. We show that, under the long-tailedness of the densities of normalized Lévy measures, the densities of infinitely divisible distributions on the half-line are subexponential if and only if the densities of their normalized Lévy measures are subexponential. Moreover, we prove that, under a certain continuity assumption, the densities of infinitely divisible distributions on the half-line are subexponential if and only if their normalized Lévy measures are locally subexponential. MSC: 60E07, 60G51 Keywords: subexponential density, local subexponentiality, infinite divisibility, Lévy measure
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Introduction and main results
The subexponentiality of infinitely divisible distributions on the half-line was characterized by Embrechts et al. [5] and on the real line by Pakes [16,17] and Watanabe [25]. The subexponentiality of a density is a stronger and more difficult property than the subexponentiality of a distribution. Some infinitely divisible distributions on the half-line such as Pareto, lognormal, and Weibull (with parameter less than 1) distributions have subexponential densities. Watanabe and Yamamuro [29] proved that the density of a self-decomposable distribution on the real line is subexponential if and only if the density on (1, ∞) of its normalized Lévy measure is subexponential. The purpose of this paper is to characterize the subexponential densities of absolutely continuous infinitely divisible distributions on the half-line under some additional assumptions. In what follows, we denote by R the real line and by R+ the half-line [0, ∞). We denote by N the set of positive integers. The symbol δa (dx) stands for the delta measure at a ∈ R. Let η and ρ be finite measures on R. We denote by η ∗ ρ the convolution of η and ρ and by ρn∗ the nth convolution power of ρ with the understanding that ρ0∗ (dx) = δ0 (dx). The characteristic function of a distribution ρ is denoted by ρ, that is, for z ∈ R, ∞ ρ(z) := eizx ρ(dx). −∞
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T. Watanabe
Let f and g be probability density functions on R. We denote by f ⊗ g the convolution of f and g and by f n⊗ the nth convolution power of f for n ∈ N. For positive functions f1 and g1 on [A, ∞) for some A ∈ R, we define the relation f1 (x) ∼ g1 (x) by limx→∞ f1 (x)/g1 (x) = 1. We use the symbols L and S in the sense of long-tailedness and subexponentiality, respectively. The subscripts d, ac, and loc mean density, absolutely continuity, and locality, respectively D EFINITION 1. (i) A nonnegative and eventually positive measurable function g on R belongs to the class L if g(x + a) ∼ g(x) for every a ∈ R. (ii) A probability density function g on R belongs to the c
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