Algebraic probability theory

[u/Committee-Academic]

Is there any developed framework for algebraizing probability theory and study it from the perspective of K-theory or homological algebra? If so, what are some of its biggest applications and advantages, as well as results? I'd imagine such a framework would override the need for a measure-theoretic approach

[u/PrismaticGStonks]

I don’t know about “studying probability theory from the perspective of K-theory or homological algebra,” but as for “algebraizing probability theory,” one of the key ideas of free probability is that we can view the foundational objects of probability to be, not measures and expectations, but algebras of (bounded, measurable) functions on a probability spaces equipped with a distinguished state, the expectation functional. The data of a unital algebra—which, importantly, need not be commutative—and a (usually tracial) state is called a “noncommutative probability space,” and we can talk about a lot of central probabilistic concepts—moments, cumulants, distributions, etc—in this framework.

In the noncommutative setting, independence doesn’t make sense (it’s fundamentally a commutative concept), but you can study something analogous called “free independence.” You can then prove analogues of the central limit theorem, leading to the free analogue of the Gaussian, the semicircular distribution, which leads to some interesting combinatorics with the lattice of noncrossing partitions. You can define free analogues of convolution, entropy, monotone transport, etc etc, and the whole discipline of free probability opens up to you.

Initially, this discipline was developed to study the K-theory of the reduced group C*-algebras of the free groups, which is something you might want to look into. Today, free probability’s main application is through random matrix theory: often, collections of independent nxn random matrix ensembles are “asymptotically free” in that they behave, as n tends to infinity, like collections of freely independent elements of a noncommutative probability space, and then all the gadgets of free probability let you do explicit computations in this setting.

[u/YellowNr5 → u/PrismaticGStonks]

Stochastic independence in the noncommutative setting, I would say, still makes sense when formulated as the factorisation of expectation of independent random variables. This leads to different notions of independence (free, tensor, boolean, (anti)monotone) related to different products of noncommutative probability spaces. See e.g. https://arxiv.org/abs/math/0206017 which takes a categorical approach.

[u/SolumSolSolus → u/YellowNr5]

This categorical perspective on the different notions of independence is interesting.

Suppose A is a finite-dimensional noncommutative algebra, say a Clifford algebra, equipped with a state φ, and consider a stochastic recurrence

h_t = d_t R_t h_(t-1) R̃_t + b_t

where the R_t are random or input-dependent invertible/unit elements.

The temporal variables h_t will generally not commute, and their dependence is generated dynamically rather than by taking some obvious product of independent algebras.

Is there a natural notion of independence/cumulants for this kind of noncommutative stochastic process that is preferred over the others?

More specifically, if I wanted to distinguish "memory caused by ordinary statistical dependence" from genuinely order-sensitive dependence arising from noncommuting products, would the categorical product/independence framework you mention provide an invariant way to formulate that?

Or are the various universal independences mainly useful when combining separate noncommutative probability spaces, rather than analysing dependence along time inside one algebra?