Noncommutative Version of The Central Limit Theorem and of Cramér’s Theorem

Abstract

Usually, a random variable f; is considered as a measurable function ξ(·) on some probability space. But a function can also be regarded as a multiplication operator, and different random variables commute as multiplication operators. Mixed moments, m(ξ₁…ξ_n),can be written as

$$m\left( {{\xi _1} \cdots {\xi _n}} \right) = \int {du 1 \cdot \xi \left( \omega \right) \ldots {\xi _n}\left( \omega \right) \cdot 1 \equiv \left\langle {{\varphi _0}, {\xi _1} \cdots {\xi _n} {\varphi _0}} \right\rangle } $$

(1.1)

where <,> denotes the scalar product in L²(μ) , ${\varphi _0}\left( \omega \right) \equiv 1$ ^(ω) =1, and the ξ_i’s are regarded as multiplication operators.