Abstract
In this paper, a subexponential density on R+∪{0} without the almost decrease is constructed. Correspondingly, a similar result is also obtained for the long-tailed density excluding the subexponential density. Additionally, in contrast to the first result, a sufficient condition is provided under which the density is almost decreasing. Some interesting conclusions are presented when the above results are applied to the study of local distribution and infinitely divisible distribution. For example, a substantial difference between the subexponential distribution and the local subexponential distribution is clearly shown.
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