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Let $(\xi_1,\eta_1)$, $(\xi_2,\eta_2),\ldots$ 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$n^{-1/2}\underset{0\leq k\leq \cdot}{\max}\,(\xi_1+\ldots+\xi_k+\eta_{k+1})$was proved under the assumption that contributions of $\underset{0\leq k\leqn}{\max}\,(\xi_1+\ldots+\xi_k)$ and $\underset{1\leq k\leq n}{\max}\,\eta_k$ tothe limit are comparable and that $n^{-1/2}(\xi_1+\ldots+\xi_{[n\cdot]})$ isattracted to a Brownian motion. In the present paper, we continue this line ofresearch and investigate a more complicated situation when$\xi_1+\ldots+\xi_{[n\cdot]}$, properly normalized without centering, isattracted to a centered stable L\'{e}vy process, a process with jumps. As aconsequence, weak convergence normally holds in the $M_1$-topology. We alsoprovide sufficient conditions for the $J_1$-convergence. For completeness, lessinteresting situations are discussed when one of the sequences $\underset{0\leqk\leq n}{\max}\,(\xi_1+\ldots+\xi_k)$ and $\underset{1\leq k\leqn}{\max}\,\eta_k$ dominates the other. An application of our main results todivergent perpetuities with positive entries is given.

Functional limit theorems for the maxima of perturbed random walks and divergent perpetuities in the $M_1$-topology