  The first passage time process of a L\'evy subordinator with heavy-tailed
L\'evy measure has long-range dependent paths. The random fluctuations that
appear under two natural schemes of summation and time scaling of such
stochastic processes are shown to converge weakly. The limit process is
fractional Brownian motion in one case and a non-Gaussian and non-stable
process in the other case. The latter appears to be of independent interest as
a random process that arises under the influence of coexisting Gaussian and
stable domains of attraction and is known from other applications to provide a
bridge between fractional Brownian motion and stable L\'evy motion.

Probability