Processes which can be represented as the maximum value of some underlying process.

An interestingly mathematically tractable way of getting interesting behaviour from boring variables, even IID ones.
The *other* mathematically convenient way of handing monotonic processes apart from branching processes and affiliated counting processes.

I had my interest in these rekindled recently by Peter Straka, after first running into them in a lecture by Paul Embrechts in terms of risk management.

My former co-supervisor Sara van de Geer then introduced another class of them to me where the maximum is not taken over the state space of a scalar random 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 dividends. Supposedly the time transform is especially rich, and the semigroup structure especially convenient? Also Meerschaert and Stoev derive some statistical properties of a class of renewal-process-based max process. (MeSt07)

Obviously this needs to be made precise, which may happen if it turns out to actually help.

## Refs

- EmKM97
- Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997) Risk Theory. In Modelling Extremal Events (pp. 21–57). Springer Berlin Heidelberg
- HeSc15
- Hees, & Scheffler. (2015) Coupled Continuous Time Random Maxima.
- Laur12
- Lauritzen, S. L.(2012) Extremal Families and Systems of Sufficient Statistics. . Springer Science & Business Media
- McFE05
- McNeil, A. J., Frey, R., & Embrechts, P. (2005) Quantitative risk management : concepts, techniques and tools. . Princeton: Princeton Univ. Press
- MeSt07
- Meerschaert, M. M., & Stoev, S. A.(2007) Extremal limit theorems for observations separated by random waiting times.
- MeSt09
- Meerschaert, M. M., & Stoev, S. A.(2009) Extremal limit theorems for observations separated by random power law waiting times.
*Journal of Statistical Planning and Inference*, 139(7), 2175–2188. DOI. - Ress91
- Ressel, P. (1991) Semigroups in Probability Theory. In H. Heyer (Ed.), Probability Measures on Groups X (pp. 337–363). Springer US DOI.
- Ress11
- 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.