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

Abstract

Let $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ 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 $n^{-1/2}\underset{0\le k\le \cdot }{\max }({\xi }_1+\dots +{\xi }_k+{\eta }_{k+1})$ was proved under the assumption that contributions of $\underset{0\le k\le n}{\max }({\xi }_1+\dots +{\xi }_k)$ and $\underset{1\le k\le n}{\max }{\eta }_k$ to the limit are comparable and that $n^{-1/2}({\xi }_1+\dots +{\xi }_{[n\cdot ]})$ is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when ${\xi }_1+\dots +{\xi }_{[n\cdot ]}$, 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 $\underset{0\le k\le n}{\max }({\xi }_1+\dots +{\xi }_k)$ and $\underset{1\le k\le n}{\max }{\eta }_k$ dominates the other. An application of our main results to divergent perpetuities with positive entries is given.

submitted for publication, 15 pages

References in corpus (1)

• A functional limit theorem for locally perturbed random walks