Algebra of Random Variables

In the algebraic axiomatization of probability theory, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to each random variable. The measurable space and the probability measure arise from the random variables and expectations by means of well-known representation theorems of analysis. One of the important features of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones.

Random variables are assumed to have the following properties:

1. complex constants are random variables;
2. the sum of two random variables is a random variable;
3. the product of two random variables is a random variable;
4. addition and multiplication of random variables are both commutative; and
5. there is a notion of conjugation of random variables, satisfying (ab)* = b*a* and a** = a for all random variables a,b and coinciding with complex conjugation if a is a constant.

This means that random variables form complex commutative *-algebras. If a = a* then the random variable a is called "real".

An expectation E on an algebra A of random variables is a normalized, positive linear functional. What this means is that

1. E(k) = k where k is a constant;
2. E(a*a) ≥ 0 for all random variables a;
3. E(a + b) = E(a) + E(b) for all random variables a and b; and
4. E(za) = zE(a) if z is a constant.

One may generalize this setup, allowing the algebra to be noncommutative. This leads to other areas of noncommutative probability such as quantum probability, random matrix theory, and free probability.