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Asymptotic Expansions for Perturbed Discrete Time Renewal Equations
Stockholm University, Faculty of Science, Department of Mathematics.
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 [en]
Renewal equation, Perturbation, Asymptotic expansion, Regenerative process, Quasi-stationary distribution, Risk process, Ruin probability
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:su:diva-95490OAI: oai:DiVA.org:su-95490DiVA: diva2:660387
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
List of papers
1. Exponential Expansions for Perturbed Discrete Time Renewal Equations
Open this publication in new window or tab >>Exponential Expansions for Perturbed Discrete Time Renewal Equations
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
Keyword
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:nbn:se:su:diva-95461 (URN)10.1002/9781118701881.ch23 (DOI)9781118539422 (ISBN)9781118701881 (ISBN)
Available from: 2013-10-29 Created: 2013-10-29 Last updated: 2016-04-28Bibliographically approved
2. Quasi-Stationary Distributions for Perturbed Discrete Time Regenerative Processes
Open this publication in new window or tab >>Quasi-Stationary Distributions for Perturbed Discrete Time Regenerative Processes
2014 (English)In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 89, 153-168 p.Article in journal (Refereed) Published
Abstract [en]

Non-linearly perturbed discrete time regenerative processes with regenerative stopping times are considered. We define the quasi-stationary distributions for such processes and present conditions for their convergence. Under some additional assumptions, the quasi-stationary distributions can be expanded in asymptotic power series with respect to the perturbation parameter. We give an explicit recurrence algorithm for calculating the coefficients in these asymptotic expansions. Applications to perturbed alternating regenerative processes with absorption and perturbed risk processes are presented.

Keyword
Regenerative process, Renewal equation, Non-linear perturbation, Quasi-stationary distribution, Asymptotic expansion, Risk process
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-95468 (URN)
Available from: 2013-10-29 Created: 2013-10-29 Last updated: 2016-04-28Bibliographically approved
3. Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes
Open this publication in new window or tab >>Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes
2014 (English)In: Modern Problems in Insurance Mathematics / [ed] Dmitrii Silvestrov, Anders Martin-Löf, Springer, 2014, 95-112 p.Chapter in book (Refereed)
Abstract [en]

We consider the problem of approximating the infinite time horizon ruin probabilities for discrete time risk processes. The approach is based on asymptotic results for non-linearly perturbed discrete time renewal equations. Under some moment conditions on the claim distributions, the approximations take the form of exponential asymptotic expansions with respect to the perturbation parameter. We show explicitly how the coefficients of these expansions can be computed as functions of the coefficients of the expansions of local characteristics for perturbed risk processes.

Place, publisher, year, edition, pages
Springer, 2014
Series
EAA Series, ISSN 1869-6929
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-95469 (URN)10.1007/978-3-319-06653-0_7 (DOI)978-3-319-06652-3 (ISBN)978-3-319-06653-0 (ISBN)
Available from: 2013-10-29 Created: 2013-10-29 Last updated: 2016-04-28Bibliographically approved

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