Generalized continuous time random walks with independent, heavy-tailed random waiting times and long range dependent jumps are considered. Their scaling limits are determined in terms of the Hermite processes and inverse of stable subordinators. These limiting processes provide an interesting new class of non-Markovian, non-Gaussian self-similar processes.<br />Tail probability estimates for the limiting process are established, which are applied in turn to establish uniform and local moduli of continuity.

Continuous time random walk; Hermite process; Strictly stable process; Waiting time process; Self-similarity; Scaling limit

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