A complementary note on the supercritical birth,death and catastrophe process

A central limit theorem for the population size of the super-critical linear birth and death process with a linear catastrophe component has previously been obtained under a fourth order moment condition on the increment distribution. In this note we show that this result is valid under a second order moment condition, and that no lesser condition will suffice. This is accomplished by giving a new, self-contained and simple proof of the asymptotic normality of a certain tail sum of independent variates.