A class of non-negative alternating regenerative processes is considered, w
here the process stays at zero random time (waiting period), then it jumps
to a random positive level and hits zero after some random period (life per
iod), depending on the evolution of the process. It is assumed that the wai
ting time and the lifetime belong to the domain of attraction of stable law
s with parameters in the interval (1/2, 1]. An integral representation for
the distribution functions of the regenerative process is obtained, using t
he spent time distributions of the corresponding alternating renewal proces
s. Given the asymptotic behaviour of the process in the regenerative cycle,
different types of limiting distributions are proved, applying some new re
sults for the corresponding renewal process and two limit theorems for the
convergence in distribution.