Based on the three-term recurrence relation of the orthogonal polynomials associated with the distribution of a random variable, the random variable as a multiplication operator is represented as the sum of three linear operators, namely the creation, annihilation and preservation operators (CAP-operators), which is called the quantum decomposition of the random variable.
This approach extends the usual Boson quantum mechanics corresponding to the Gaussian measure to the quantum mechanics associated with random variables.
In this seminar, we first discuss analytic representations of the CAP operators in terms of differential operators with polynomial coefficients.
Secondly, we introduce the notion of finite type random variable and characterize type-2 and type-3 real-valued random variables.
We also discuss a necessary condition for a random variable to be of finite.
This allows to prove that the Beta and the uniform distributions are of infinite type.