Draft:R Transform

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 $$G_{\mu }(z)=\int _{\mathbb {R} }{\frac {1}{z-t}}\,d\mu (t).$$ 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 $\{1,\ldots ,n\}$. In the convention used here, $$R_{\mu }(w)=\sum _{n=1}^{\infty }\kappa _{n}w^{n-1}.$$ In particular, $\kappa _{1}$ is the mean and $\kappa _{2}$ is the variance. Another convention uses $\sum _{n\geq 1}\kappa _{n}w^{n}=wR_{\mu }(w)$.

Additive convolution

If $\mu \boxplus \nu$ is the distribution of the sum of freely independent self-adjoint random variables with compactly supported distributions $\mu$ and $\nu$, then $$R_{\mu \boxplus \nu }(w)=R_{\mu }(w)+R_{\nu }(w)$$ 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