Transient phenomena for random walks in the absence of the expected value of jumps

Abstract

Let ξ, ξ1, ξ2, ... be independent identically distributed random variables, and Sn:=Σj=1n,ξj, \( \bar S \):= supn≥0Sn. If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution \( \bar S \) as a → 0 and \( \bar S \) tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models: 1. The first model assumes that ξj can be represented as ξj = ζj + αηj, where ζ1, ζ2, ... and η1, η2, ... are two independent sequences of independent random variables, identically distributed in each sequence, such that supn≥0Σj=1n ζj = ∞, supn≥0Σj=1nηj < ∞, and \( \bar S \) < ∞ almost surely. 2. In the second model we consider a triangular array scheme with parameter a and assume that the right tail distribution P(ξj ≥ t) ∼ V (t) as t→∞ depends weakly on a, while the left tail distribution is P(ξj 0.