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

Abstract: 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 $\underset{0\leq k\leq n}{\max}\,(\xi_1+\ldots+\xi_k)$ and $\underset{1\leq k\leq n}{\max}\,\eta_k$ to the limit are comparable and that $n^{-1/2}(\xi_1+\ldots+\xi_{[n\cdot]})$ is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when $\xi_1+\ldots+\xi_{[n\cdot]}$, properly normalized without centering, is attracted to a centered stable L\'{e}vy 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 $\underset{0\leq k\leq n}{\max}\,(\xi_1+\ldots+\xi_k)$ and $\underset{1\leq k\leq n}{\max}\,\eta_k$ dominates the other. An application of our main results to divergent perpetuities with positive entries is given.