Processes which can be represented as the maximum value of some underlying process.
An interestingly mathematically tractable way of getting interesting behaviour from borining variables, even IID ones.
The other mathematically conveneint way of handing monotonic processes apart from branching processes and affiliated counting processes.
My former co-supervisor Sara van de Geer then introduced another class of them to me where the maximum is not take ove rthstate space of a scalar ranum variable, but maximum deviation inequalities for convergence of empirical distributions; These latter ones are not so tractable, which is why I strategically retreated.
Peter assures me that if I read Ressel I will be received diviends.
Supposedly the time transform is especially rich, and the semigroup structure especially convenient?
Obviously this needs to be made precise, which may happen if it turns out to actually help.
- Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997) Risk Theory. In Modelling Extremal Events (pp. 21–57). Springer Berlin Heidelberg
- Lauritzen, S. L.(2012) Extremal Families and Systems of Sufficient Statistics. . Springer Science & Business Media
- McNeil, A. J., Frey, R., & Embrechts, P. (2005) Quantitative risk management : concepts, techniques and tools. . Princeton: Princeton Univ. Press
- Ressel, P. (1991) Semigroups in Probability Theory. In H. Heyer (Ed.), Probability Measures on Groups X (pp. 337–363). Springer US
- Ressel, P. (2011) A revision of Kimberling’s results — With an application to max-infinite divisibility of some Archimedean copulas. Statistics & Probability Letters, 81(2), 207–211. DOI.
See original: Maximum processes