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Exponential Expansions for Perturbed Discrete Time Renewal Equations
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2013 (English)In: Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference / [ed] Ilia B. Frenkel, Alex Karagrigoriou, Anatoly Lisnianski, Andre Kleyner, Chichester: John Wiley & Sons, 2013, 349-362 p.Chapter in book (Refereed)
Abstract [en]

This chapter presents results about the asymptotic behavior of the solution x(φ)(n) of a perturbed discrete time renewal equation as φ--> 0 and n-->? simultaneously. It consider two cases of so-called quasi-stationary and pseudo-stationary asymptotics, where the limiting distribution f (0)(k) may be, respectively, improper or proper. The author improves the asymptotic relation to the much more advanced form of an exponential asymptotic expansion. The chapter illustrates theoretical results by examples related to queuing systems and risk processes. It briefly shows the way of getting the renewal equation. It repeats the method of finding a similar continuous time renewal equation for ruin probabilities, given, for example in Feller (1966) and Grandell (1991).

Place, publisher, year, edition, pages
Chichester: John Wiley & Sons, 2013. 349-362 p.
Keyword [en]
perturbed discrete time renewal, pseudo-stationary asymptotics, quasi-stationary asymptotics, queuing systems, risk processes
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-95461DOI: 10.1002/9781118701881.ch23ISBN: 9781118539422 (print)ISBN: 9781118701881 (print)OAI: oai:DiVA.org:su-95461DiVA: diva2:660252
Available from: 2013-10-29 Created: 2013-10-29 Last updated: 2016-04-28Bibliographically approved
In thesis
1. Asymptotic Expansions for Perturbed Discrete Time Renewal Equations
Open this publication in new window or tab >>Asymptotic Expansions for Perturbed Discrete Time Renewal Equations
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study the asymptotic behaviour of the solution of a discrete time renewal equation depending on a small perturbation parameter. In particular, we construct asymptotic expansions for the solution of the renewal equation and related quantities. The results are applied to studies of quasi-stationary phenomena for regenerative processes and asymptotics of ruin probabilities for a discrete time analogue of the Cramér-Lundberg risk model.

Place, publisher, year, edition, pages
Department of Mathematics, 2013. 19 p.
Keyword
Renewal equation, Perturbation, Asymptotic expansion, Regenerative process, Quasi-stationary distribution, Risk process, Ruin probability
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-95490 (URN)
Presentation
2013-11-20, 306, Matematiska institutionen, Hus 6, Kräftriket, Stockholm, 15:15 (English)
Opponent
Supervisors
Available from: 2013-11-06 Created: 2013-10-29 Last updated: 2013-11-06Bibliographically approved
2. Perturbed discrete time stochastic models
Open this publication in new window or tab >>Perturbed discrete time stochastic models
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, nonlinearly perturbed stochastic models in discrete time are considered. We give algorithms for construction of asymptotic expansions with respect to the perturbation parameter for various quantities of interest. In particular, asymptotic expansions are given for solutions of renewal equations, quasi-stationary distributions for semi-Markov processes, and ruin probabilities for risk processes.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2016. 48 p.
Keyword
Renewal equation, Perturbation, Asymptotic expansion, Regenerative process, Risk process, Semi-Markov process, Markov chain, Quasi-stationary distribution, Ruin probability, First hitting time, Solidarity property
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-128979 (URN)978-91-7649-422-6 (ISBN)
Public defence
2016-06-02, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript.

Available from: 2016-05-10 Created: 2016-04-11 Last updated: 2017-11-14Bibliographically approved

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