Using Boolean cumulants to study multiplication and anti-commutators of free random variables

Abstract

We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the * -distribution of the product of two selfadjoint freely independent random variables, and in connection to the distribution of the anticommutator of such random variables.

Principal angles between random subspaces and polynomials in two free projections

“…Example 2 (Anticommutator of free projections). The free anticommutator pq + qp has attracted some attention in the recent years [3]. While one may repeat the argument given in the proof of Theorem 4, it is actually simpler to observe that pq + qp can be written as (p + q) 2 − (p + q).…”

On the anti-commutator of two free random variables

“…Equation (2) takes a simpler form when a and b have the same distribution, which means that ϕpa n q " ϕpb n q for all n P N, or equivalently that κ n paq " κ n pbq for all n P N. This phenomenon was also observed in [FMNS20], for the formula in terms of Boolean cumulants.…”

“…More recently, a formula to compute the Boolean cumulants of the free anti-commutator ab `ba in terms of the individual Boolean cumulants of a and b was provided in [FMNS20]. On the other hand, Ejsmont and Lehner studied quadratic forms of even variables [EL20b] (see also [EL17]) and the limiting distribution that arises from sums of commutators and anti-commutators [EL20a].…”

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

“…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.…”

“…Powerful as it is, subordination itself does not allow to prove all the results which we are studying here. We take advantage of connections between free probability and Boolean cumulants established recently in [10,12]. We develop ideas from [12], in particular we provide a new expansion of the reciprocal of the additive subordination function in terms of Boolean cumulants…”