Convolution and limit theorems for conditionally free random variables

Abstract

We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its R-transform. We calculate explicitly the distributions of the conditionally free Gaussian and

On multiplicative conditionally free convolution

“…Moreover, as shown in [8], Theorem 2.2, if A 1 , A 2 are two unital * -algebras, each A k endowed with two states (normalized, positive, linear functionals) ψ k , ϕ k (k = 1, 2), then there exist two unique states ψ, ϕ on A 1 * A 2 (free product of * -algebras with amalgamation over the unit) with respect to which A 1 and A 2 are c-free and ψ |A k = ψ k , ϕ |A k = ϕ k .…”

“…This paper presents some results in conditionally free (or, shorter, following [8], c-free) probability. The concept of c-freeness was developed in the early 1990s (see [6], [7], [8]) in connection to freeness with amalgamation.…”

“…The concept of c-freeness was developed in the early 1990s (see [6], [7], [8]) in connection to freeness with amalgamation. Throughout the paper we will use the definition of c-freeness from [7] and [8].…”

“…The concept of c-freeness was developed in the early 1990s (see [6], [7], [8]) in connection to freeness with amalgamation. Throughout the paper we will use the definition of c-freeness from [7] and [8]. Namely, if A is an algebra and ϕ, ψ : A −→ C are two normalized linear functionals, then the family {A j } j∈J of subalgebras of A is said to be c-free if (i) ψ(a 1 · · · a n ) = 0, (ii) ϕ(a 1 · · · a n ) = ϕ(a 1 ) · · · ϕ(a n ) for all a j ∈ A ε(j) with ε(j) ∈ J such that ε(k) = ε(k + 1), 1 ≤ k ≤ n − 1, and ψ(a j ) = 0, 1 ≤ j ≤ n.…”

“…Those are power series with the property that if X and Y are free random variables, then R X+Y = R X + R Y and S XY = S X S Y . While a c-free version of the R-transform is constructed in [8] and used in several papers (such as [24], [17], [14]) for the study of c-free additive convolution of measures or of c-freeness with amalgamation, the literature lacked a similar development for the multiplicative case.…”