Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T17:54:41.993Z Has data issue: false hasContentIssue false
  • English
  • Français

On randomly spaced observations and continuous-time random walks

Part of: Limit theorems Stochastic processes

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: [email protected]
** Postal address: Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, Zagreb, Croatia.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Anderson, K. K. (1987).Limit theorems for general shock models with infinite mean intershock times.J. Appl. Prob. 24,449–456.CrossRefGoogle Scholar
[2] Barczyk, A. and Kern, P. (2013).Scaling limits of coupled continuous time random walks and residual order statistics through marked point processes.Stoch. Process. Appl. 123,796–812.CrossRefGoogle Scholar
[3] Basrak, B. (2015).Limits of renewal processes and Pitman–Yor distribution.Electron. Commun. Prob. 20.Google Scholar
[4] Basrak, B. and Špoljarić, D. (2015).Extremes of random variables observed in renewal times.Statist. Prob. Lett. 97,216–221.CrossRefGoogle Scholar
[5] Berman, S. (1962).Limiting distribution of the maximum term in sequences of dependent random variables.Ann. Math. Statist. 33,894–908.Google Scholar
[6] Csáki, E. and Hu, Y. (2003).Lengths and heights of random walk excursions. InDiscrete Random Walks, (Paris, 2003),Association of Discrete Mathematics and Theoretical Computer Science,Nancy,45–52 Google Scholar
[7] De Haan, L. and Ferreira, A. (2006).Extreme Value Theory: An Introduction.Springer ,New York.CrossRefGoogle Scholar
[8] De Haan, L. and Resnick, S. I. (1977).Limit theory for multivariate sample extremes.Z. Wahrscheinlichkeitsth. 40,317–337.Google Scholar
[9] Durrett, R. (2010).Probability: Theory And Examples, 4th edn.Cambridge University Press.CrossRefGoogle Scholar
[10] Embrechts, P.,Klüppelberg, and Mikosch, T. (1997).Modelling Extremal Events: For Insurance and Finance (Appl. Math. (New York) 33).Springer,Berlin.Google Scholar
[11] Faÿ, G.,González-Arévalo, B. Mikosch, T. and Samorodnitsky, G. (2006).Modeling teletraffic arrivals by a Poisson cluster process.Queueing Systems 54,121–140.CrossRefGoogle Scholar
[12] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.Google Scholar
[13] Godrèche, C.,Majumdar, S. N. and Schehr, G. (2015).Statistics of the longest interval in renewal processes.J. Statist. Mech. Theory Exp. 2015,PO3014.Google Scholar
[14] Lamperti, J. (1961).A contribution to renewal theory.Proc. Amer. Math. Soc. 12,724–731.CrossRefGoogle Scholar
[15] Leadbetter, M. R.,Lindgren, G. and Rootzén, H. (1983).Extremes and Related Properties of Random Sequences and Processes Springer ,New York.CrossRefGoogle Scholar
[16] Meerschaert, M. M. and Scheffler, H.-P. (2004).Limit theorems for continuous-time random walks with infinite mean waiting times.J. Appl. Prob. 41,623–638.CrossRefGoogle Scholar
[17] Meerschaert, M. M. and Stoev, S. A. (2009).Extremal limit theorems for observations separated by random power law waiting times.J. Statist. Planning Inference 139,2175–2188.Google Scholar
[18] Montroll, E. W. and Weiss, G. H. (1965).Random walks on lattices. II..J. Math. Phys. 6,167–181.Google Scholar
[19] Pancheva, E.,Mitov, I. K. and Mitov, K. V. (2009).Limit theorems for extremal processes generated by a point process with correlated time and space components.Statist. Prob. Lett. 79,390–395.Google Scholar
[20] Perman, M. (1993).Order statistics for jumps of normalised subordinators.Stoch. Process. Appl. 46,267–281.Google Scholar
[21] Pitman, J. and Yor, M. (1997).The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator.Ann. Prob. 25,855–900.Google Scholar
[22] Resnick, S. I. (1987).Extreme Values, Regular Variation and Point Processes.Springer,New York.Google Scholar
[23] Resnick, S. I. (2007).Heavy-Tail Phenomena.Springer,New York.Google Scholar
[24] Schumer, R.,Baeumer, B. and Meerschaert, M. M. (2011).Extremal behavior of a coupled continuous time random walk.Physica A 390,505–511.CrossRefGoogle Scholar
[25] Seneta, E. (1976).Regularly Varying Functions (Lecture Notes Math. 508).Springer,Berlin.Google Scholar
[26] Shanthikumar, J. G. and Sumita, U. (1983).General shock models associated with correlated renewal sequences.J. Appl. Prob. 20,600–614.Google Scholar