On the Distribution of Claims for a Process Terminating on a Run of Critical Events
Abstract
In modern actuarial risk management, the temporal clustering of severe claims is as critical as their cumulative financial magnitude. This study investigates a stochastic risk process that terminates upon the occurrence of a run of consecutive claims exceeding a predefined critical threshold. First, using recursive conditioning techniques, we derive the exact moment generating function for the total severity of exceedances accumulated prior to termination, providing an explicit probability density function for the exponential case when . Second, we determine the exact cumulative distribution and probability mass functions for the maximum number of consecutive non-exceedances observed between two critical claims via first-order linear difference equations. The derived analytical expressions bridge the theory of runs and practical risk management, offering direct tools for dynamic solvency monitoring and operational stress testing without relying on asymptotic approximations
Keywords
References
- [1] Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., & Nesbitt, C. J. (1987). Actuarial mathematics-actuarial mathematics by bowers, hickman, gerber, jones and nesbitt [published in 1986 by the society of actuaries]. Transactions of the Faculty of Actuaries, 41, 91-94. https://doi.org/10.1017/S0071368600009812
- [2] Klugman, S. A., Panjer, H. H., & Willmot, G. E. (2012). Loss models: from data to decisions. John Wiley & Sons 715.
- [3] Koutras, M. V. (1996). On a waiting time distribution in a sequence of Bernoulli trials. Annals of the Institute of Statistical Mathematics, 48(4), 789-806. https://doi.org/10.1007/ BF00052333
- [4] Willmot, G. E., & Lin, X. S. (2001). Lundberg approximations for compound distributions with insurance applications. Springer Science & Business Media 156.
- [5] Asmussen, S., Albrecher, H. (2010). Ruin Probabilities. World Scientific 14.
- [6] Koutras, V. M., & Koutras, M. V. (2020). Exact distribution of random order statistics and applications in risk management. Methodology and Computing in Applied Probability, 22(4), 1539-1558. https://doi.org/10.1007/ s11009-018-9662-z
- [7] Koutras, V. M., Koutras, M. V., & Dafnis, S. D. (2022). A Family of Induced Distributions. Methodology and Computing in Applied Probability, 24(3), 1833–1848. https://doi.org/10.1007/s11009-021-09887-1
- [8] Eryilmaz, S. (2016). A new class of lifetime distributions. Statistics & Probability Letters, 112, 63-71. https://doi.org/10.1016/j.spl.2016.01.023
Details
Primary Language
English
Subjects
Statistics (Other)
Journal Section
Research Article
Publication Date
April 29, 2026
Submission Date
November 14, 2025
Acceptance Date
April 13, 2026
Published in Issue
Year 2026 Volume: 47 Number: 2