[Paper Review] On multiplicative conditionally free convolution

Mihai Popa, Jiun-Chau Wang|ArXiv.org|May 2, 2008

Mathematical Analysis and Transform Methods12 references3 citations

TL;DR

This paper introduces a conditionally free analogue of the S-transform, termed the ${}^{c}T$-transform, using non-crossing linked partitions to characterize multiplicative conditionally free convolution. It establishes the multiplicative property $T_{XY} = T_X T_Y$, enabling analytical descriptions of infinite divisibility and limit theorems in conditionally free probability, extending free probability tools to the multiplicative, c-free setting via combinatorial enumeration.

ABSTRACT

Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The result is applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility.

Motivation & Objective

• To develop a multiplicative analogue of the S-transform in the context of conditionally free (c-free) probability, where such a tool was previously lacking.
• To establish the multiplicative property of the ${}^{c}T$-transform, $T_{XY} = T_X T_Y$, for conditionally free random variables.
• To apply the ${}^{c}T$-transform to characterize infinite divisibility and study limit distributions in the multiplicative c-free convolution framework.
• To provide explicit combinatorial formulas for the $T$- and ${}^{c}T$-transforms using non-crossing linked partitions, generalizing results from [9].
• To bridge the gap between free probability and conditionally free probability by extending the $S$-transform machinery to the multiplicative, c-free setting.

Proposed method

• The ${}^{c}T$-transform is constructed via combinatorial enumeration of non-crossing linked partitions (NCL), generalizing the approach in [15] and [9].
• The method relies on the recurrence relations for c-free cumulants ${}^{c}R^n$ and their generating functions, derived from the moment-cumulant relations involving $\varphi$ and $\psi$.
• The key identity $^{c}R(z[1+m_X(z)]) (1+M_X(z)) = M_X(z)(1+m_X(z))$ links the $^{c}R$-transform to moments under $\varphi$ and $\psi$, enabling the definition of the $^{c}T$-transform.
• The $^{c}T$-transform is defined as the inverse of the $S$-transform, with $T_X(z) = \left(\frac{1}{z} R(z)\right) \circ R^{\langle-1\rangle}(z)$, ensuring multiplicative behavior under c-free independence.
• Explicit formulas for the coefficients of $T_X$ and ${}^{c}T_X$ are derived using the structure of non-crossing linked partitions, with $m_n = \sum_{\gamma \in NCL(n)} t_0^{n-|\gamma|} \prod_{B \in \gamma} t_{|B|-1}$ and $M_n = \sum_{\gamma \in NCL(n)} t_0^{n-|\gamma|} \prod_{B \in ext(\gamma)} {}^{c}t_{|B|-1} \prod_{B \in int(\gamma)} t_{|B|-1}$.
• The proof uses induction on $n$, decomposing partitions via their first block and constructing recursive relations based on singly- and doubly-covered elements.

Experimental results

Research questions

• RQ1 Can a multiplicative analogue of the S-transform be constructed in the conditionally free probability setting?
• RQ2 Does the resulting $^{c}T$-transform satisfy the multiplicative property $T_{XY} = T_X T_Y$ for c-free random variables?
• RQ3 Can the $^{c}T$-transform be used to characterize infinite divisibility in the multiplicative c-free convolution framework?
• RQ4 What is the combinatorial structure underlying the moment-cumulant relations in the c-free setting, and how can it be encoded via non-crossing linked partitions?
• RQ5 How do the moment-generating series $m_X(z)$ and $M_X(z)$ relate to the $^{c}T$-transform under c-free independence?

Key findings

• The ${}^{c}T$-transform is defined as the inverse of the $S$-transform in the c-free setting, ensuring $T_{XY} = T_X T_Y$ for c-free $X$ and $Y$, generalizing the free case.
• The moment $m_n$ of a random variable under $\psi$ is given by $m_n = \sum_{\gamma \in NCL(n)} t_0^{n-|\gamma|} \prod_{B \in \gamma} t_{|B|-1}$, where $t_k$ are coefficients of the $T$-transform.
• The moment $M_n$ under $\varphi$ is expressed as $M_n = \sum_{\gamma \in NCL(n)} t_0^{n-|\gamma|} \prod_{B \in ext(\gamma)} {}^{c}t_{|B|-1} \prod_{B \in int(\gamma)} t_{|B|-1}$, with $^{c}t_k$ encoding c-free cumulants.
• The recursive structure of non-crossing linked partitions allows the derivation of the $^{c}T$-transform via induction, using block decomposition and coverage types (singly/doubly covered).
• The $^{c}T$-transform enables analytical characterization of infinite divisibility and limit theorems in multiplicative c-free convolution, extending tools from free probability.
• The paper provides a complete combinatorial formula for the $^{c}T$-transform coefficients using non-crossing linked partitions, generalizing earlier results from [9] to the c-free case.