R Transform

Short description: Transform that linearizes free additive convolution

In free probability, the R-transform is a transform of a probability distribution that converts free additive convolution into addition. Introduced by Dan-Virgil Voiculescu, it describes the distribution of a sum of freely independent random variables. Its power-series coefficients are the free cumulants.

Definition

Let $\mu$ be a compactly supported probability measure on $\mathbb{ℝ}$, with Cauchy transform $$G_{\mu }(z)=\underset{\mathbb{ℝ}}{\int }\frac{1}{z-t}d\mu (t).$$ Near infinity, $G_{\mu }(z)=z^{-1}+O(z^{-2})$. Its compositional inverse branch $K_{\mu }$ satisfies $K_{\mu }(w)=w^{-1}+O(1)$ near zero. The R-transform is $$R_{\mu }(w)=K_{\mu }(w)-\frac{1}{w}.$$ The singularity at zero is removable. The inverse is local, rather than a global inverse of $G_{\mu }$.

Free cumulants

Writing $m_n=\int t^{n}d\mu (t)$, the free cumulants ${\kappa }_n$ are determined by $$m_n=\sum _{\pi \in NC(n)}\prod _{B\in \pi }{\kappa }_{|B|},$$ where $NC(n)$ denotes the noncrossing partitions of $\{1,\dots ,n\}$. In the convention used here, $$R_{\mu }(w)=\sum _{n=1}^{\mathrm{\infty }}{\kappa }_n{w}^{n-1}.$$ In particular, ${\kappa }_1$ is the mean and ${\kappa }_2$ is the variance. Another convention uses $\sum _{n\ge 1}{\kappa }_n{w}^n=w{R}_{\mu }(w)$.

Additive convolution

If $\mu ⊞\nu$ is the distribution of the sum of freely independent self-adjoint random variables with compactly supported distributions $\mu$ and $\nu$, then $$R_{\mu ⊞\nu }(w)=R_{\mu }(w)+R_{\nu }(w)$$ for sufficiently small $w$. This follows from the vanishing of mixed free cumulants.

Examples

With the convention above, standard examples are:

Distribution	R-transform
Point mass ${\delta }_a$	$R(w)=a$
Semicircular distribution with mean $m$ and variance ${\sigma }^2$	$R(w)=m+{\sigma }^{2}w$
Free Poisson distribution with rate $\lambda >0$ and jump size $\alpha >0$	$R(w)=\frac{\lambda \alpha }{1-\alpha w}$

The free Poisson convention in this table has ${\kappa }_n=\lambda {\alpha }^n$.

See also

• Free probability
• Free convolution
• Marchenko–Pastur distribution

External links

• Speicher's lecture notes: arXiv record (full text)
• Nica–Speicher book: authors' page with downloadable text and errata