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On randomly spaced observations and continuous-time random walks

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  24 October 2016

Bojan Basrak  and
Drago Špoljarić
Bojan Basrak*
Affiliation:
University of Zagreb
Drago Špoljarić*
Affiliation:
University of Zagreb
*
* Postal address: Department of Mathematics, University of Zagreb, Bijenićka 30, Zagreb, Croatia. Email address: bbasrak@math.hr
** Postal address: Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, Zagreb, Croatia.
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Abstract

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.

Keywords

Extreme value theorypoint processrenewal processcontinuous-time random walkexcursionsojourn time

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F17: Functional limit theorems; invariance principles 60G55: Point processes 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 888 - 898
Copyright
Copyright © Applied Probability Trust 2016 

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