Let (ξ1, η1), (ξ2, η2),… be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J1-topology on the Skorokhod space of n − 1 / 2 max 0 ≤ k ≤ [ n ⋅ ] ( ξ 1 + … + ξ k + η k + 1 ) $n^{-1/2}\underset {0\leq k\leq [n\cdot ]}{\max }\,(\xi _{1}+\ldots +\xi _{k}+\eta _{k+1})$ was proved under the assumption that contributions of max 0 ≤ k ≤ n ( ξ 1 + … + ξ k ) $\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})$ and max 1 ≤ k ≤ n η k to the limit are comparable and that n−1/2(ξ1+… + ξ[n⋅]) is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when ξ1+… + ξ[n⋅], properly normalized without centering, is attracted to a centered stable Lévy process, a process with jumps. As a consequence, weak convergence normally holds in the M1-topology. We also provide sufficient conditions for the J1-convergence. For completeness, less interesting situations are discussed when one of the sequences max 0 ≤ k ≤ n ( ξ 1 + … + ξ k ) $\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})$ and max 1 ≤ k ≤ n η k dominates the other. An application of our main results to divergent perpetuities with positive entries is given.