Functional limit theorems for the maxima of perturbed random walk and divergent perpetuities in the M (1)-topology

摘要

Let (xi (1), eta (1)), (xi (2), eta (2)),aEuro broken vertical bar 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 was proved under the assumption that contributions of and to the limit are comparable and that n (-1/2)(xi (1)+aEuro broken vertical bar + xi ([nai...])) 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)+aEuro broken vertical bar + xi ([nai...]), properly normalized without centering, is attracted to a centered stable L,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 and dominates the other. An application of our main results to divergent perpetuities with positive entries is given.