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Scaling limit results for the sum of many inverse Lévy subordinators

The first passage time process of a Lévy subordinator with heavy-tailed Lévy 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 Lévy motion.