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Abstract: In this paper, we put forward and deal with three problems related to local long-tailed distributions and local subexponential distributions. We prove that there exists a local long-tailed distribution that is not locally subexponential, and also present some other relations and differences between the local distribution class and the global distributions class. [Copyright &y& Elsevier]

Abstract: Let be a proper distribution on or and be a non-negative integer-valued random variable with masses . Denote . The main result of this paper is that under some suitable conditions, belongs to the convolution equivalent distribution class if and only if belongs to the convolution equivalent distribution class. As applications, some known results on random sums have been extended and improved, which give a positive answer under certain conditions to Problem 1 of . Similarly, some corresponding results for the local distributions and densities have been obtained. [Copyright &y& Elsevier]