  This paper is a continuation of work arXiv:2006.09583 devoted to
establishment of the convergence rate in the strong invariance principle for
cumulative processes. We establish optimal rate of convergence for the case
when regeneration periods and increments over them have only power moments of
order greater than 2. Under this power-type conditions two types of
approximation by Wiener process are proved: the rate of convergence in the
Strassen's invariance principle and inequalities for the probability that
random process deviates from the approximating one.
