Conditional expectations through Boolean cumulants and subordination – towards a better understanding of the Lukacs property in free probability

Abstract

We use here a recent idea of studying functions of free random variables using Boolean cumulants. We develop idea of explicit calculations of conditional expectation using Boolean cumulants. We demonstrate Boolean cumulants approach allows to calculate explicitly some conditional expectations of functions in free random variables. We present how Boolean cumulants together with subordination simplify proofs of some results which are known in research literature.2010 Mathematics Subject Classification. Primary: 46L54. Secondary: 62E10.

The Matsumoto-Yor property in free probability via subordination and Boolean cumulants

“…One of the implicit results of this paper is a methodological remark, that Boolean cumulants prove to be useful tool when dealing with conditional expectation of expressions involving free random variables. It confirms the observation already noted in [18] in context of regression versions of the Lukacs property.…”

“…We will need two formulas involving Boolean cumulants. They can be found in [10] and [12] and were used also in [18]. Proposition 2.6.…”

“…for f, g : A → A, which seem to be important in relations between subordination and Boolean cumulants. In ( [18], Proposition 3.4) a general and rather complicated formula expressing η f Y and η f,g Y in terms of η Y was proved for f and g being analytic functions in the unit disc. Consequently, in a special case of 0 ≤ Y < I and…”

The Matsumoto-Yor property in free probability via subordination and Boolean cumulants

“…One of the implicit results of this paper is a methodological remark, that Boolean cumulants prove to be a useful tool when dealing with conditional expectations of expressions involving free random variables. It confirms the observation already noted in Szpojankowski and Wesołowski (2020) in the context of regression versions of the Lukacs property.…”

“…We will need two formulas involving Boolean cumulants. They can be found in Fevrier et al (2020) and Lehner and Szpojankowski (2021) and were used also in Szpojankowski and Wesołowski (2020).…”