Lower limits and upper limits for tails of random sums supported on

Based on Rudin (1973), Foss and Korshunov (2007) and Denisov et al. (2008), we study the lower limits and upper limits of the quotients of tails as x-->[infinity], where [tau] is a non-negative integer-valued random variable and F 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 Watanabe (2008) under certain conditions. In addition, we give some corresponding results for the local versions and density versions.