On the almost decrease of a subexponential density

Abstract

For a subexponential density, so far, there has been no positive conclusion or counter example to show whether it is almost decreasing. In this paper, a subexponential density supported on R + ∪ {0} without the almost decrease is constructed by a little skillful method. The density is a positive piecewise linear function with a more normal shape. Correspondingly, there exists a local subexponential distribution which is not locally almost decreasing. Based on an example of Cline [8], some similar results are also obtained for the long-tailed density excluding the subexponential density and the local long-tailed distribution excluding the local subexponential distribution. Finally, the paper shows that, for the local subexponentiality of a distribution supported on R, the local almost decreasing condition is necessary in some sense.

Second Order Subexponentiality And infinite divisibility

“…Next we prove (iii). As in (ii), we have q(x) ∼ p(x) ∼ x −2 l(x), ν(x) ∼ μ(x) ∼ x −1 l(x), and Hence, we get (1-12) by (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11). Next we prove (iv).…”

“…as x → ∞ and then A → ∞. Thus, from (3-8)- (3)(4)(5)(6)(7)(8)(9)(10)(11) and the assumption, we obtain that…”

Subexponential densities of infinitely divisible distributions on the half line

“…Next, we characterize the case where ν (1) is absolutely continuous with a long-tailed density. Note that Corollary 1.1 of Jiang et al [5] is analogous to Theorem 1.1 below but their proof is not valid.…”

Second order subexponentiality and infinite divisibility

“…An exponential moment assumption in the above theorem is necessary for the restriction of the class S loc in the two sided case. See Jian et al [10] for the detailed account.…”