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Abstract: Based on , and , we study the lower limits and upper limits of the quotients of tails as , where is a non-negative integer-valued random variable and is a distribution supported on . Some of the new results, which are different from the corresponding results of , give a positive answer to Problem 2 of under certain conditions. In addition, we give some corresponding results for the local versions and density versions. [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]