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

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 F∗τ¯(x)F¯(x) as x→∞x→∞, where ττ is a non-negative integer-valued random variable and FF is a distribution supported on R. Some of the new results, which are different from the corresponding results of R+, 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.