Multilinear function series and transforms in free probability theory

Abstract

The algebra Mul [[B]] of formal multilinear function series over an algebra B and its quotient SymMul [[B]] are introduced, as well as corresponding operations of formal composition. In the setting of Mul [[B]], the unsymmetrized R-and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R-and T-transforms (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.

On multiplicative conditionally free convolution

“…Suppose that the B μ n converge uniformly on the compact subsets of D to the function B μ . Then (9) shows that the weak limit μ of {μ n } ∞ n=1 is uniquely determined and hence the full sequence μ n must converge to μ. Conversely, let us suppose that the sequence μ n converges weakly to μ. Since {B μ n } ∞ n=1 is a normal family of analytic functions, it has a subsequence that converges uniformly on the compact subsets of D to an analytic function B.…”

“…By virtue of (9), it suffices to show that the function B μ is a constant function. Set c = T ζ dμ(ζ).…”

“…The notion of non-crossing linked partitions, that we will largely use in this section, is a generalization of non-crossing partitions. It was first discussed in [9], in connection with the T -transform. By a non-crossing linked partition γ of the ordered set {1, 2, .…”

“…Applying the induction hypothesis, the above equality becomes Example: The partition σ = (1, 4, 5, 9), (2, 3), (5, 6, 7), (8), (9,10) from NCL 1 (10) is represented graphically below:…”

“…The present paper shows the construction of a suitable c-free version of the Stransform (in fact, as in [9], a more natural choice is its multiplicative inverse, called the T -transform) and demonstrates some of its applications in limit theorems and characterization of infinite divisibility.…”