Algebraic probability

If you liked it then you prob’ly put a ring on it

Commonly used algebraic structures over probability, as seen in, for example, Free probability.

Algebraic probability

In algebraic probability we do not take the Kolmogorov axioms as foundational, but do away with measure theory and event spaces, starting rather from RVs and expectations.

George Lowther introduces this and a connection to quantum probablility in characteristically plain-talk style 1, 2, which is one useful generalization. We can also get a handle on “non-commutative” probability this way, and are especially interested in free probability in that context. But my knowledge is exhausted now. If you wish to know more, here are some people who actually know stuff about

Group structures which arise in classic probability

There is obviously a lot going on . But I do not know it. See, however, John Baez’s category theory lists.


Aji, S.M., and R.J. McEliece. 2000. The Generalized Distributive Law.” IEEE Transactions on Information Theory 46 (2): 325–43.
Almost, Chris. 2011. “Semimartingales and Stochastic Integration,” 73.
Applebaum, David. 2009. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.
Ressel, Paul. 1991. Semigroups in Probability Theory.” In Probability Measures on Groups X, edited by Herbert Heyer, 337–63. Springer US.
———. 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.
Ruzsa, Imre, and Gábor J Székely. 1988. Algebraic Probability Theory. John Wiley & Sons Inc.
Speicher, Roland. 2019. Lecture Notes on ‘Free Probability Theory’.” arXiv:1908.08125 [Math], August.
Xia, Xiang-Gen. 2019. A Simple Introduction to Free Probability Theory and Its Application to Random Matrices.” arXiv:1902.10763 [Math], February.

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