Extension of the invariance principle for compound renewal processes to the zones of moderately large and small deviations

The invariance principle for compound renewal processes is extended (in the sense of asymptotic equivalence) to the zone of moderately large and small deviations. It is assumed that the vector $(\tau,\zeta)$, which “governs” the process, satisfies certain moment conditions (for example, the Cramér condition), and its components $\tau$ and $\zeta$ are either independent or linearly dependent. This extension holds, in particular, for random walks.