A note on randomly stopped sums with zero mean increments.

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum S_ν = X₁ + ··· + X_ν of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of {X₁,X₂, . . .}. The conditions are provided for the relation P(S_ν > x) ∼ Eν P(X₁ > x) to hold, as x →∞, involving the finiteness of E|X₁|. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment E|X₁|^r for some r > 1 was assumed. [ABSTRACT FROM AUTHOR]