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On randomly spaced observations and continuous-time random walks. (English) Zbl 1351.60066

Summary: We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy-tailed steps, the limiting behavior of extreme observations until a given time \(t\) tends to be rather involved. We describe the asymptotics and extend several partial results which appeared in this setting. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous-time random walks.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60F17 Functional limit theorems; invariance principles
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems