On polynomial recurrence property of "Markov-up" processes

This work is a continuation of [Kalikaeva, MPRF, 23(2):225-240]. The object of study is ``Markov-up processes’’ on ${\mathbb{Z}}_{+}$ and the moment of downcrossing a certain barrier. The processes considered in this paper differ from Markov ones by the presence of a memory in certain parts of the trajectory. In our previous paper [Kalikaeva, MPRF, 23(2):225-240] exponential recurrence conditions were established. In this paper polynomial recurrence properties are considered under certain new assumptions.