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. Another 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. He points out that is this arrives from another interesting algebraic structure over distributions not derived from variable summation.
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.
Embrechts, Paul, Claudia Klüppelberg, and Thomas Mikosch. 1997. “Risk Theory.” In Modelling Extremal Events, 21–57. Applications of Mathematics 33. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-642-33483-2_2.
Hees, and Scheffler. 2015. “Coupled Continuous Time Random Maxima.”
Lauritzen, Steffen L. 2012. Extremal Families and Systems of Sufficient Statistics. Springer.
McNeil, Alexander J, Rüdiger Frey, and Paul Embrechts. 2005. Quantitative Risk Management : Concepts, Techniques and Tools. Princeton: Princeton Univ. Press.
Meerschaert, Mark M., and STILIAN A. Stoev. 2007. “Extremal Limit Theorems for Observations Separated by Random Waiting Times.” https://www.researchgate.net/profile/Stilian_Stoev2/publication/222433383_Extremal_limit_theorems_for_observations_separated_by_random_power_law_waiting_times/links/0deec5177f56d29898000000.pdf.
Meerschaert, Mark M., and Stilian A. Stoev. 2009. “Extremal Limit Theorems for Observations Separated by Random Power Law Waiting Times.” Journal of Statistical Planning and Inference 139 (7): 2175–88. https://doi.org/10.1016/j.jspi.2008.10.005.
Ressel, Paul. 1991. “Semigroups in Probability Theory.” In Probability Measures on Groups X, edited by Herbert Heyer, 337–63. Springer US. https://doi.org/10.1007/978-1-4899-2364-6_26.
———. 2011. “A Revision of Kimberling’s Results — with an Application to Max-Infinite Divisibility of Some Archimedean Copulas.” Statistics & Probability Letters 81 (2): 207–11. https://doi.org/10.1016/j.spl.2010.11.008.