Let (ξ 1 , η 1 ), (ξ 2 , η 2 ), . . . be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J 1 -topology on the Skorokhod space of n −1/2 max 0≤k≤· (ξ 1 +. . .+ξ k +η k+1 ) was proved under the assumption that contributions of max 0≤k≤n (ξ 1 +. . .+ξ k ) and max 1≤k≤n η k to the limit are comparable and that n −1/2 (ξ 1 +. . .+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 ξ 1 + . . . + 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 M 1 -topology. We also provide sufficient conditions for the J 1 -convergence. For completeness, less interesting situations are discussed when one of the sequences max 0≤k≤n (ξ 1 + . . . + ξ 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.