Maximum processes

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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.

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 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: The Living Thing / Notebooks Maximum processes