Draft:R-Transform

In free probability, the R-transform is an analytical tool used to characterize and study the distributions of non-commutative random variables. It serves as the free analogue of the cumulant generating function from classical probability, playing a role analogous to the logarithm of the Fourier transform. Its usefulness lies in simplifying the computation of the free additive convolution of two distributions due to its nice additivity property: for freely independent random variables $x$ and $y$, the R-transform satisfies $R_{x+y}(z)=R_{x}(z)+R_{y}(z)$.

Definition

Let $(A,\phi )$ be a non-commutative probability space, i.e. a unital algebra $A$ over $\mathbb {C}$ equipped with a unital linear functional $\phi :A\to \mathbb {C}$. For $a\in A$ we define a's R-transform as a formal power series:

$$R_{a}(z):=\sum _{n\in \mathbb {N} }\kappa _{n}z^{n-1},$$

where $\kappa _{n}$ represent $a$'s free cumulants. $R_{a}$ indeed characterizes $a$'s (non-commutative) distribution: $a$'s distribution is given (by definition) by its moments $m_{n}:=\phi (a^{n}),n\in \mathbb {N}$, or, equivalently, its (formal) moment series $M(z):=1+\sum _{n\in \mathbb {N} }m_{n}z^{n}$. By the moment-cumulant formula, $a$'s moments uniquely determine $a$'s free cumulants $\kappa _{n}$ and vice versa, meaning $a$'s distribution can be given by the collection $\{\kappa _{n}\}_{n\in \mathbb {N} }$, or, equivalently, a formal series such as $R_{a}(z):=\sum _{n\in \mathbb {N} }\kappa _{n}z^{n-1}$.

Analyticity

Under certain conditions, the R-transform can be shown to be analytic on a domain. The following discussion will be limited to the case of an element $a$ in a *-probability space $(A,\phi )$, i.e. a non-commutative probability space such that $A$ is a *-algebra and $\phi$ is positive ($\forall a\in A$ $\phi (a^{*}a)\geq 0$), where $a$ has the additional constraints of being self-adjoint with exponentially bounded moments: $\exists M>0|\forall n\in \mathbb {N}$ $m_{n}:=\phi (a^{n})\leq M^{n}$. One can show that for such $a$ there is a uniquely determined compactly supported probability measure $\mu \in {\mathcal {P}}(\mathbb {R} )$ on $\mathbb {R}$ with moments $m_{n}$:

$$(1)\int _{\mathbb {R} }t^{n}d\mu (t)=\phi (a^{n})\,\forall n\in \mathbb {N} ,$$ $$(2)\,supp(\mu )\in [-M,M],$$

meaning $a$'s distribution can be given by the measure $\mu$, an analytic object. In particular, $\mu$ has compact support, meaning its Cauchy transform (or minus the Stieltjes transform) $G_{\mu }(z):=\int _{\mathbb {R} }{\frac {1}{z-t}}d\mu (t),\,z\in \mathbb {C} ^{+}$ can be expressed as a convergent power series in a neighborhood of infinity:

$$\forall |z|>M,\,G_{\mu }(z)=\sum _{n\in \mathbb {N} _{0}}{\frac {m_{n}}{z^{n+1}}}={\frac {1}{z}}M\left({\frac {1}{z}}\right),$$

where $M(z):=1+\sum _{n\in \mathbb {N} }m_{n}z^{n}$ is $a$'s moment series, which, by this equation, is seen to be analytic in a neighborhood of 0. Using the moment-cumulant formula., one can show $C(zM(z))=M(z)$ where $C(z):=1+\sum _{n\in \mathbb {N} }\kappa _{n}z^{n}$ is $a$'s cumulant series. By these series relations we obtain $C(G(z))=C\left({\frac {1}{z}}M\left({\frac {1}{z}}\right)\right)=M\left({\frac {1}{z}}\right)=zG(z)$, meaning $K(z):={\frac {C(z)}{z}}$ satisfies $K(G(z))=z$ and thus also $G(K(z))$ (at least formally). Writing $K(z)={\frac {1}{z}}+R_{a}(z)$ links the (formally defined) R-transform $R_{a}(z):=\sum _{n\in \mathbb {N} }\kappa _{n}z^{n-1}$ to the analytic $G_{\mu }$. One can indeed show, using Rouché's theorem, that $R_{a}$ has an analytic series expansion in a neighborhood of $0$

Application to Free Convolution

For a compactly supported measure $\mu \in {\mathcal {P}}(\mathbb {R} )$, we can take the polynomial algebra $A:=\mathbb {C} [X]$, and define an involution $X^{*}=X$, as well as a unital linear functional:

$$\phi (p(X)):=\int _{\mathbb {R} }p(t)d\mu (t)$$

for every polynomial $p$. $\phi$ is clearly positive, thus, $X$ is a self-adjoint element of a $*$-probability space with exponentially bounded moments. Thus, for every compactly supported probability measure $\mu$ on $\mathbb {R}$ there is a self-adjoint element in a $*$-probability space, such that the measure satisfying (1) and (2) at the beginning of the section coincides with $\mu$. It is noteworthy that, as a result, one can go from self-adjoint elements to compactly supported probability measures and vice versa. In particular, for two such measures $\mu ,\nu$, we can find a $*$-probability space $(A,\phi )$ with free self-adjoint elements $x,y$ corresponding to $\mu ,\nu$, respectively (for instance, one can use the method above to obtain $*$-probability spaces and self-adjoints within them corresponding to $\mu ,\nu$, then one may simply take $A$ to be the unital algebra free product of these spaces). Using freeness, it is easy to show that $x+y$'s moments are exponentially bounded, meaning there exists a unique corresponding compactly supported probability measure $\mu \boxplus \nu$, the measure obtained by the free convolution of $\mu ,\nu$. Note that this measure does not depend on the choice of $A,\phi ,x,y$.

For free elements $x,y$, we have that their mixed cumulants vanish, yielding the most useful property of the R-transform:

$$R_{x+y}(z)=R_{x}(z)+R_{y}(z),$$ since $R_{x+y}$'s coefficients are just $\kappa _{n}^{x+y}=\kappa _{n}^{x}+\kappa _{n}^{y}$. Using this property and the connection to the Cauchy transform, the following recipe for $\mu \boxplus \nu$ is obtained:

• Calculate the Cauchy transforms $G_{\mu },G_{\nu }$
• Solve for $K_{\mu },K_{\nu }$ (using $G_{(\cdot )}(K_{(\cdot )}(z))=z$)
• Solve for $R_{\mu },R_{\nu }$ (using $K_{(\cdot )}(z)=1/z+R_{(\cdot )}(z)$)
• Additivity of cumulants yields $R_{\mu \boxplus \nu }=R_{\mu }+R_{\nu }$
• Reverse the process to obtain $G_{\mu \boxplus \nu }$
• Apply the Stieltjes inversion formula

Whereby in the last bullet point, note that a probability measure $\gamma$ on $\mathbb {R}$ satisfies the assumptions to guarantee the Stieltjes inversion:

$$\gamma ((a,b))+{\frac {\gamma (\{a,b\})}{2}}=-\lim _{y\rightarrow 0^{+}}{\frac {1}{\pi }}\int _{a}^{b}Im(G_{\gamma }(x+iy))dx,\,\forall a,b\in \mathbb {R} ,a<b$$

General Case

This discussion can be extended to a more general setting (for instance, $a$ with a corresponding measure of unbounded support; see, for instance).

Example

Using the recipe from the section above, $\mu \boxplus \nu$ may be calculated for for $\mu =\nu ={\frac {\delta _{-1}+\delta _{1}}{2}}$. First, the Cauchy transform can be evaluated as $G(z)={\frac {z}{z^{2}-1}}$. Using the second step, one obtains $K(z)={\frac {1\pm {\sqrt {1+4z^{2}}}}{2z}}$. Subtracting $1/z$ yields $R(z)={\frac {{\sqrt {1+4z^{2}}}-1}{2z}}$ where one notes $R(0)=\kappa _{1}=\mu _{1}=\int xd\mu =1/2-1/2=0$ allows for the rejection of the negative square root solution $R(z)={\frac {-{\sqrt {1+4z^{2}}}-1}{2z}}$. Multiply by two to obtain $R_{\mu \boxplus \nu }$, add $1/z$ to obtain $K_{\mu \boxplus \nu }(z)$, and use $z=K_{\mu \boxplus \nu }(G_{\mu \boxplus \nu }(z))$ to conclude $G_{\mu \boxplus \nu }(z)=(z^{2}-4)^{-{\frac {1}{2}}}$. Apply the Stieltjes Inversion Formula to see that $\mu \boxplus \nu$ is a "semicircular distribution" (actually the standard semicircular distribution, characterized by having variance 1) with density: $${\begin{cases}{\frac {1}{\pi {\sqrt {4-t^{2}}}}}&|t|\leq 2\\0&|t|>2\end{cases}}$$

In particular, from the example above, one observes that the free convolution of two discrete distributions yields a continuous one. If one "translates" this statement into the classical setting—there exist independent discrete random variables (finite range), whose sum yields a continuous random variable distributed according to the standard Gaussian (uncountable range)—one soon sees the infeasibility in the classical case (note that the semicircular distribution is in many ways the "free analog" of the classical Gaussian distribution). This can be viewed as an instance of the stark difference between commutative and non-commutative probability settings.

References

• Voiculescu, Dan V.; Dykema, Kenneth J.; Nica, Alexandru (1992). Free Random Variables. American Mathematical Society. ISBN 978-0-8218-1140-5.