Given a supercritical Galton‒Watson process {Z n } and a positive sequence {εn }, we study the limiting behaviors of ℙ(S Z n /Zn ≥εn ) with sums S n of independent and identically distributed random variables X i and m=𝔼[Z 1]. We assume that we are in the Schröder case with 𝔼Z 1 log Z 1<∞ and X 1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z 1 is subexponentially distributed, we further obtain the convergence rate of Z n+1/Z n to m as n→∞.