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

Abstract

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.

A renewal theorem and supremum of a perturbed random walk

“…In the most interesting, third case, that is, if P(η > x) ∼ cx −2 for some c > 0, both random walk and the perturbation have comparable contributions, see [18,9] along with generalization to functional limit theorems. Further developments have been made recently in [10], where the assumption Eξ 2 k < ∞ is dispensed with.…”

“…and so the tail is essentially bigger than that of B, since L(0, x)/L(x) → ∞ as x → ∞; see (10). Appearance of the function L is probably the most interesting phenomenon here.…”

Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

“…A survey of various results for the so defined PRWs can be found in the book [13]. An incomplete list of more recent papers addressing various aspects of the PRWs includes [1,9,16,21,22,23].…”

A result on power moments of Lévy-type perpetuities and its application to the Lp -convergence of Biggins’ martingales in branching Lévy processes

“…A major part of the recent book [23] is concerned with the so defined perturbed random walks, both multiplicative and additive. We refer to the cited book for numerous applications of these random sequences and to [1,15,17,24,25] for more recent contributions.…”