Generalised operations in free harmonic analysis

Abstract

In this article we study the formal side of operations in free harmonic analysis and examine the emerging general picture of all this. We establish an analytic correspondence of semi-rings between Witt vectors and free probability, by building on previous joint work with Friedrich and McKay (Formal groups, Witt vectors and free probability, 2012. arXiv:1204.6522). In particular, an exponential map, which relates the free additive convolution semigroup on ${\mathbb {R}}$ with the free multiplicative convolution semigroup on either the unit circle or the positive real axis of compactly supported, freely infinitely divisible probability measures, is derived with complex analytic methods. Then we define several novel operations on these sets, discuss their relation with classically infinitely divisible measures and determine the internal geometry of the spaces involved. Finally, we formalise the structure induced by the various operations we have introduced, in the language of operads and algebraic theories.

1 Introduction

The theory of probability measures on metric groups is a well-studied subject, cf. e.g. [22]. A fundamental theorem therein states that the set of probability measures M(X) on a group X, which is also a separable metric space, becomes a topological semi-group for the operation $*$, which is defined as the convolution of measures, i.e. for $\mu ,\nu \in M(X)$, one has

$$\begin{aligned} \mu *\nu (A):=\int _X\mu (Ax^{-1})d\nu (x),\quad A\in {\mathcal {B}}(X), \end{aligned}$$

with ${\mathcal {B}}(X)$ the Borel $\sigma$-algebra on X. An important observation is that the restriction imposed by positivity, reduces the original structure on the underlying space to a substructure on the associated space of measure, i.e.

$$\begin{aligned} \text {group }\rightarrow \text { semigroup}. \end{aligned}$$

An analogous theory, where the notion of independence is replaced by freeness, was introduced and developed by Voiculescu [24,25,26], Voiculescu and Bercovici [4, 5], and Bercovici and Pata [3], who related free harmonic analysis with classically infinitely divisible probability measures, as an isomorphism of semigroups.

Bercovici and Voiculescu [5] discussed more general convolution operations on the set of probability measures supported on ${\mathbb R}$. Namely, they considered selfadjoint polynomial expressions $Q(T_1,\dots ,T_n)$ in n non-commuting variables $T_i$, and showed that the resulting distribution $\mu$,

$$\begin{aligned} \mu :=Q^{\square }(\mu _1,\dots ,\mu _n), \end{aligned}$$

with $\mu _i=\mu _{T_i}$, $i=1,\dots ,n$, for free random variables $T_1,\dots ,T_n$ in a $W^*$-probability space $(A,\tau )$, can be expressed by the $\mu _i$ only.

The generalisation of this line of reasoning is Male’s Traffic [19], who considers the operad generated by graph polynomials with an appropriate notion of independence, called Traffic freeness.

Here we shall consider more general operations directly on proper subsets of the probability measures supported on ${\mathbb R}$, i.e. we have

$$\begin{aligned} Q^{\square }(\mu _1,\dots ,\mu _n)\rightarrow {\mathcal {P}}_n(\mu _1,\dots ,\mu _n), \end{aligned}$$

where the ${\mathcal {P}}_n$ either define a ${\mathcal {P}}$-operad or the operations of an algebraic theory.

Our motivation to consider directly such general operations on probability distributions comes from the relation between free harmonic analysis, complex cobordism and the ring of Witt vectors which has been previously found by McKay and the author [8, 9].

This link permitted us [9], amongst other things, to derive a formula which expresses Voiculescu’s free multiplication [25] in terms of his free addition [24], but also to define an additional convolution operation on distributions which, a priori, is not induced by the underlying algebra structure. Subsequently, it was shown [10] that for several classes of probability distributions, this new operation indeed respects the analytic constraints given by positive definiteness, and that it induces a partial ring structure.

Here we generalise the initial analytic results in [10] to the complete set of freely infinitely divisible probability measures with compact support and introduce further operations, several of them in analogy with the classical ones for Witt vectors [14]. Then we formalise the (partial) algebra of these operations in more basic categorical and algebraic terms.

A fundamental tool in our approach are the free cumulants, which have been introduced by Speicher [23] and which form the base of the combinatorial approach to free probability, as developed by Nica and Speicher [21].

From a general perspective, understanding the set, respectively algebra of operations, although not traditional in probability theory, has been of primordial importance in other mathematical fields and hence intensively studied. For example, in formal language theory [6] several algebraic operations have been described which preserve the set of recognisable formal languages, e.g. the shuffle product, or in the theory of Witt vectors one has the functorial endomorphisms, e.g. Frobenius and Verschiebung.

Our results appear not only to be important from a general perspective but should also have fruitful possible applications, e.g. in random matrix theory.

Organisation of the paper. In Sect. 1, after giving the general motivation, the needed concepts from free probability theory are recalled. Section 2, develops the analytic counterpart of the algebraic theory presented in [9] for freely infinitely divisible probability measures with compact support. The first subsection discusses the holomorphic linearisation of the S-transform [25]. As next, before the analytic Witt semi-ring structure is established, an adapted characterisation of infinitely divisible measures in terms of cumulants is proved. Then, several endomorphisms of the space of infinitely divisible measures are introduced and the geometric meaning of the Lévy-Khintchine representation discussed. In the next subsection, the relation with classically infinitely divisible probability measures is established via the Bercovici–Pata bijection [3]. Finally, we close Sect. 2, by discussing infinitesimal limits. In the last part, Sect. 3, which consists of three subsections, we introduce a categorical perspective. First, we discuss convex sets and the barycentric algebra in general terms, and its relation to the operations previously defined on infinitely divisible probability measures. In particular we point out the fact that one cannot naturally mix two different co-ordinate systems, i.e. operations based on both moments and cumulants. Then we introduce the fundamental (finite) Giry monad and its relations to infinitely divisible measures. In the last part, we summarise the analytic operations established in the previous sections within the language of operads and the more general framework of algebraic theories.

1.1 Preliminaries from free probability

Basic information about free probability can be obtained from the monographs [21, 26]. Here we shall consider the one-dimensional case only. Let k be a field of characteristic zero, $\mathbf {Alg}_k$ the category of associative and unital k-algebras, and $\mathbf {cAlg}_k$ the associative, commutative and unital k-algebras.

Definition 1

Let $A\in \mathbf {Alg}_k$, with unit $1_A$, and let $\phi :A\rightarrow k$ be a k-linear functional with $\phi (1_A)=1_k$. The pair $(A,\phi )$, is called a non-commutative k-probability space, and the elements $a\in A$ the random variables.

The algebraic k-probability spaces form a category, which we denote by $\mathbf {Alg_kP}$. The subcategory of commutative algebraic probability spaces is denoted by $\mathbf {cAlg_kP}$. The full subcategory of $C^*$-probability spaces $\mathbf {C^*Alg_{{\mathbb C}}P}$, has as objects pairs $(A,\phi )$, where A is a $C^*$-algebra and $\phi$ a state.

Definition 2

The law or distribution of a random variable $a\in A$, is the k-linear functional $\mu _a\in k[[x]]$, given by $\mu _a(x^n):=\phi (a^n)$, $n\in {\mathbb N}$. The coefficients $(m_n(a))_{n\in {\mathbb N}}$ of the corresponding power series $\mu _a$ are the moments of a.

Definition 3

(Freeness) Let $(A,\phi )\in \mathbf {Alg_kP}$ and I an index set. A family of sub-algebras $(A_i)_{i\in I}$, with $1_A\in A_i\subset A$, $i\in I$, is called freely independent if for all $n\in {\mathbb N}^*$, and $a_k\in {A_{i_k}\cap \ker (\phi )}$, $1\le k\le n$, also

$$\begin{aligned} a_1\cdots a_n\in {\ker (\phi )}, \end{aligned}$$

holds, given that $i_k\ne i_{k+1}$, $1\le k< n$.

One should note that the requirement for freeness is that consecutive indices must be distinct, thus $i_j=i_{j+2}$ is possible.

Let $M_c:\mathbf {Top}_{\hbox {lcH}}\rightarrow \mathbf {Set}$ denote the functor from the category of locally compact Hausdorff topological spaces to the category of sets, which assigns to every $T\in \mathbf {Top}_{\hbox {lcH}}$ the set of compactly supported Borel probability measures on T. We note that $\mathbf {Top}_{\hbox {lcH}}$ is dual to the category of commutative $C^*$-algebras.

Hence, $M_c({\mathbb C})$ corresponds to the set of all compactly supported Borel probability measures on ${\mathbb C}$, and similarly $M_c({\mathbb R}), M_c({\mathbb R}^*_+)$ and $M_c(S^1)$ to the subsets of compactly supported probability measures on ${\mathbb R}$, ${\mathbb R}^*_+:=(0,\infty )$ and the unit circle $S^1$, respectively. Let $M^*(-)\subset M_c(-)$ be the set of compactly supported probability measures with non-vanishing first moment and finally, let $\hbox {Meas}$ and $\hbox {Meas}_{c}$ denote the functors from $\mathbf {Top}_{\hbox {lcH}}$ to the finite and compactly supported finite Borel measures, respectively.

If a is a self-adjoint element in a $C^*$-probability space, and therefore satisfies $a=a^*$, then there exists a unique, compactly supported probability measure $\mu _a\in M_c({\mathbb R})$, such that for all $n\in {\mathbb N}$,

$$\begin{aligned} \int _{{\mathbb R}} x^n d\mu _a(x)=\phi (a^n) \end{aligned}$$

holds.

Let us recall from [26, pp. 21,30] the notion of the free additive and free multiplicative convolution.

For free random variables $a,b\in (A,\phi )$, with distributions $\mu _a$ and $\mu _b$, the distributions $\mu _{a+b}$ and $\mu _{ab}$ depend only on $\mu _a$ and $\mu _b$. They are called the additive free and multiplicative free convolution of $\mu _a$ and $\mu _b$, respectively, and are denoted by $\mu _a\boxplus \mu _b$ and $\mu _a\boxtimes \mu _b$.

Definition 4

Let $\mu \in M_{c}({\mathbb R})$. If for every $n\in {\mathbb N}^*$, there exists a $\mu _n\in M_c({\mathbb R})$, which satisfies

$$\begin{aligned} \mu =\underbrace{\mu _n\boxplus \dots \boxplus \mu _n}_{n \text { summands}} \end{aligned}$$

then $\mu$ is called $\boxplus$-infinitely divisible. Let $\mu \in M_c({\mathbb R}^*_+)$ or $\mu \in M^*(S^1)$. If for every $n\in {\mathbb N}^*$ there exists a $\mu _n\in M_c({\mathbb R}^*_+)$ or $\mu \in M^*(S^1)$ which satisfies

$$\begin{aligned} \mu =\underbrace{\mu _n\boxtimes \dots \boxtimes \mu _n}_{n \text { factors}} \end{aligned}$$

then $\mu$ is called $\boxtimes$-infinitely divisible.

Following the notation used in [2, 7], we denote by $\hbox {ID}_c({\mathbb R},\boxplus )$ the set of all $\boxplus$-infinitely divisible probability measures on ${\mathbb R}$ with compact support, and analogously $\hbox {ID}_c({\mathbb R}_+^*,\boxtimes )$ the set of all compactly supported probability measures on ${\mathbb R}_+^*$ which are $\boxtimes$-infinitely divisible and $\hbox {ID}^*(S^1,\boxtimes )$ the set of all $\boxtimes$-infinitely divisible probability measures on $S^1$ with non-vanishing first moment.

For the necessary definitions and properties concerning Voiculescu’s R- and S-transform one should consult [24,25,26] and for its combinatorial description [21]. Let us point out that we shall, by abuse of notation, consider R either as a map into k[[z]] or zk[[z]], which however should be clear from the context.

2 Free harmonic analysis and Witt vectors

In [9, 11], we derived a relation between the one-dimensional free additive and the one-dimensional free multiplicative convolution, which permits to completely understand $\boxtimes$ in terms of $\boxplus$ algebraically. In concrete terms the relation is

$$\begin{aligned} {\text {LOG}}(a\boxtimes b)={\text {LOG}}(a)\boxplus {\text {LOG}}(b), \end{aligned}$$

where

$$\begin{aligned} {\text {LOG}}(a):=R^{-1}\left( \frac{d}{dz}\ln S(a)\right) . \end{aligned}$$

However, in higher dimensions such a relation does not hold in general, as we showed in [10, 11].

Somewhat earlier, Mastnak and Nica [20] showed, using Hopf algebraic methods and in particular characters, that in the one-dimensional case, the S-transform can be algebraically linearised. Subsequently, Cébron [7] established a semi-group homomorphism for freely infinitely divisible probability distributions by applying a Lévy-Khintchine type representation, and related the classical with the free case via the Bercovici–Pata bijection. Anshelevich and Arizmendi [1] then discussed a map, the wrapping transformationW, which is a morphism of semigroups between the set of probability measures on the real line and the unit circle, with $\boxplus$ and $\boxtimes$ as convolution operation, respectively, and showed that W transforms the additive free, Boolean, and monotone convolution into the respective multiplicative convolutions.

Now, by using the complex analytic description of compactly supported, freely infinitely divisible probability measures, we shall give first a novel analytic derivation of such an exponential morphism and then introduce some new analytic operations on measures itself.

2.1 Holomorphic linearisation

The derivations in this subsection are based on the analytic characterisations given in [26, Section 3.7]. As usual, $\mathbb H$ denotes the upper half-plane, ${\overline{\mathbb H}}$ the closed upper half-plane, $-{\overline{\mathbb H}}$ the closed lower half-plane, $\mathfrak {R}(z)$ and $\mathfrak {I}(z)$ denotes the real and imaginary part of a complex number z, respectively.

We consider the following sets of germs of analytic functions, cf. [26, Theorems 3.7.2, 3.7.3 and 3.7.4.]:

• (1) Let ${\mathcal {R}}:={\mathcal {R}}_{\infty ,\boxplus }({\mathbb R})$ denote the set of stalks of functions, R(z), which are analytic in a neighbourhood of $({\mathbb C}\setminus {\mathbb R})\cup \{0\}$ (direct limit), and satisfy for all $z\in \mathbb H$:

  $$\begin{aligned} R({\bar{z}})=\overline{R(z)}\quad \text {and}\quad \mathfrak {I}(R(z))\ge 0. \end{aligned}$$

  (1)

• (2) Let ${\mathcal {V}}:={\mathcal {V}}_{\infty ,\boxtimes }({\mathbb R}^*_+)$ denote the set of stalks of functions, v(z), which are analytic in a neighbourhood of $({\mathbb C}\setminus {\mathbb R})\cup [-1,0]$ (direct limit), and satisfy for all $z\in \mathbb H$:

  $$\begin{aligned} v({\bar{z}})=\overline{v(z)}\quad \text {and}\quad \mathfrak {I}(v(z))\le 0. \end{aligned}$$

• (3) Let ${\mathcal {U}}:={\mathcal {U}}_{\infty ,\boxtimes }(S^1)$ denote the sheaf of functions, u(z), which are analytic in the shifted half-plane ${H}_{-1/2}:=\{z\in {\mathbb C}~|\mathfrak {R}(z)>-\frac{1}{2}\}$ and satisfy for all $z\in H_{-1/2}$:

  $$\begin{aligned} \mathfrak {R}(u(z))\ge 0. \end{aligned}$$

Lemma 1

The following properties hold:

• (1) $({\mathcal {R}},+)$, $({\mathcal {V}},+)$ and $({\mathcal {U}},+)$ are commutative monoids with neutral element 0.

• (2) The sets ${\mathcal {R}}$, ${\mathcal {V}}$ and ${\mathcal {U}}$ are convex cones, i.e. if $f,g\in {\mathcal {R}}, {\mathcal {V}}, {\mathcal {U}}$ and $\alpha ,\beta \in {\mathbb R}_+$, then $\alpha f+\beta g\in {\mathcal {R}}, {\mathcal {V}}, {\mathcal {U}}$.

Proof

We only have to show the second statement, as the first one follows from $\alpha =\beta =1$ and $\alpha =0$ or $\beta =0$, respectively.

Let us consider the claim for the set ${\mathcal {R}}$ first. If $f(z),g(z)\in {\mathcal {R}}$ and $\alpha ,\beta \ge 0$ then $\alpha f({\overline{z}})=\overline{\alpha f(z)}$ and $\mathfrak {I}(\alpha f(z))=\alpha \mathfrak {I}(f(z))\ge 0$ for $z\in \mathbb H$. Further, $\alpha f({\overline{z}})+\beta g({\overline{z}})=\overline{\alpha f(z)}+\overline{\beta g(z)}$ and $\mathfrak {I}(\alpha f(z)+\beta g(z))=\alpha \mathfrak {I}(f(z))+\beta \mathfrak {I}(g(z))\ge 0$.

The remaining cases for the sets ${\mathcal {V}}$ and ${\mathcal {U}}$ are treated similarly. $\square$

The following two statements are now consequences of [26, Theorems 3.7.2, 3.7.3 and 3.7.4].

Lemma 2

$(\hbox {ID}_c({\mathbb R},\boxplus ),\boxplus )$ is a commutative monoid with neutral element $\delta _0$. Further, it is a sub-monoid of $(M_c({\mathbb R}),\boxplus )$ and the R-transform is an isomorphism of commutative monoids:

$$\begin{aligned} R:({\text {ID}}_c({\mathbb R},\boxplus ),\boxplus )\rightarrow ({\mathcal {R}}_{\infty ,\boxplus }({\mathbb R}),+). \end{aligned}$$

Proof

The first statement follows from

$$\begin{aligned} \mu \boxplus \nu =\mu _n^{\boxplus n}\boxplus \nu _n^{\boxplus n}=(\underbrace{\mu _n\boxplus \nu _n}_{\in M_c({\mathbb R})})^{\boxplus n} \end{aligned}$$

for some $\mu _n,\nu _n\in M_c({\mathbb R})$, $n\in {\mathbb N}^*$. $\square$

Lemma 3

(Holomorphic linearisation of $\boxtimes$) We have:

• (1) There exists an isomorphism of commutative semi-groups,

  $$\begin{aligned} {\text {EXP}}:({\mathcal {V}},+)\rightarrow ({\text {ID}}_c({\mathbb R}_+^*,\boxtimes ),\boxtimes ), \end{aligned}$$

  given by $v(z)\mapsto S^{-1}(e^{v(z)})$, which linearises $\boxtimes$.

• (2) There exists an isomorphism of commutative semi-groups,

  $$\begin{aligned} {\text {EXP}}:({\mathcal {U}},+)\rightarrow ({\text {ID}}^*(S^1,\boxtimes ),\boxtimes ), \end{aligned}$$

  given by $u(z)\mapsto S^{-1}(e^{u(z)})$, which linearises $\boxtimes$.

Proof

It follows from the following equalities:

$$\begin{aligned} S_{\mu \boxtimes \nu }(z)=S_{\mu }\cdot S_{\nu }=e^{v(z)}\cdot e^{u(z)}=e^{v(z)+u(z)}. \end{aligned}$$

$\square$

As next we shall relate the free additive convolution on ${\mathbb R}$ with the free multiplicative convolution on the unit circle $S^1$.

Proposition 4

There exists a monomorphism of commutative semi-groups

$$\begin{aligned} {\text {EXP}}:({\text {ID}}_c({\mathbb R},\boxplus ),\boxplus )\rightarrow ({\text {ID}}^*(S^1,\boxtimes ),\boxtimes ) \end{aligned}$$

given by

$$\begin{aligned} \mu \mapsto S^{-1}(e^{-iR_{\mu }(i(z+1/2))}). \end{aligned}$$

Proof

Let $\varphi :H_{-1/2}\rightarrow \mathbb H$ be the conformal map $z\mapsto \varphi (z):=i(z+\frac{1}{2})$. It induces an injective map ${\hat{\varphi }}:{\mathcal {R}}\rightarrow {\mathcal {U}}$, which associates to $R_{\mu }\in {\mathcal {R}}_{\infty ,\boxplus }({\mathbb R})$, the holomorphic map

$$\begin{aligned} u(z):=-iR_{\mu }\left( i(z+\frac{1}{2})\right) . \end{aligned}$$

Indeed, $u(z)\in {\mathcal {U}}$, as for $\mathfrak {R}(z)>-\frac{1}{2}$ we have $\mathfrak {I}(i(z+\frac{1}{2}))>0$, and so, by assumption, $\mathfrak {I}(R_{\mu }(i(z+\frac{1}{2})))\ge 0$, from which we obtain that

$$\begin{aligned} \mathfrak {R}(-iR_{\mu }\left( i\left( z+\frac{1}{2}\right) \right) )=\mathfrak {I}(R_{\mu }\left( i\left( z+\frac{1}{2}\right) \right) )\ge 0. \end{aligned}$$

Now, from [26, Theorem 3.7.4] the first claim follows.

For $\delta _0$ we have $R_{\delta _0}(z)\equiv 0$ which is mapped onto 1 and hence corresponds to $\delta _{e^0}$. Next, for $\mu ,\nu \in M_{\infty ,\boxplus }({\mathbb R})$ we have $R_{\mu }+R_{\nu }=R_{\mu \boxplus \nu }$, and further

$$\begin{aligned}&\left( -iR_{\mu }\left( i\left( z+\frac{1}{2}\right) \right) \right) +\left( -iR_{\nu }\left( i\left( z+\frac{1}{2}\right) \right) \right) \\&\quad =-i(R_{\mu }+R_{\nu })\circ \varphi (z)=-i(R_{\mu \boxplus \nu })\circ \varphi (z), \end{aligned}$$

from which the property of a semi-group morphism follows. $\square$

We have the following examples:

• For $R(z)=bz$, i.e. the semi-circular law of radius $2\sqrt{b}$ and centred at the origin, we get $z\mapsto e^{bz+\frac{b}{2}}$, i.e. the free unitary Brownian motion.

• For $R(z)=a$, i.e. the Dirac measure $\delta _a$ at a, we get $a\mapsto e^{-ia}$.

The free Poisson distribution (fP) with rate $\lambda \ge 0$ and jump size $\alpha \in {\mathbb R}$, cf. e.g. ([21, Prop. 12.11], or [26, p. 34]), is the limit in distribution for $N\rightarrow \infty$ of

$$\begin{aligned} \nu _{N,\lambda ,\alpha }:=\left( \left( 1-\frac{\lambda }{N}\right) \delta _0+\frac{\lambda }{N}\delta _{\alpha }\right) ^{\boxplus N}, \end{aligned}$$

with $\boxplus N$ in the exponent denoting the N-fold free additive self-convolution.

The R-transform of the limit $\nu _{\infty ,\lambda ,\alpha }:=\lim _{N\rightarrow \infty }\nu _{N,\lambda ,\alpha }$ is

$$\begin{aligned} {R}_{\nu _{\infty ,\lambda ,\alpha }}(z)=\lambda \alpha \frac{1}{1-\alpha z}=\sum _{n=0}^{\infty }\lambda \alpha ^{n+1}z^n=\lambda \alpha +\lambda \alpha ^2 z+\lambda \alpha ^3z^2+\cdots . \end{aligned}$$

(2)

Let ${\mathcal {R}}_{[-1,0]}({\mathbb R})\subset {\mathcal {R}}_{\infty ,\boxplus }({\mathbb R})$ be the subset of stalks of R-transforms which are also analytic around $[-1,0]$ (direct limit set). Then ${\mathcal {R}}_{[-1,0]}({\mathbb R})$ is not empty, as it contains, e.g. the

• Dirac delta $\delta _a$, $a\in {\mathbb R}$,

• (Centred) semi-circular $a+bz$, $a\in {\mathbb R}$, $b>0$,

• Free Poisson $\lambda \alpha /(1-\alpha z)$, $\lambda \in {\mathbb R}_+$ and $|\alpha |<1$.

The next result seems not to have been considered in the literature so far. Namely, we shall relate the free additive convolution on ${\mathbb R}$ with the free multiplicative convolution on ${\mathbb R}_+^*$. However, the complex analytic machinery for compactly supported and freely infinitely divisible measures, only provides a statement which applies to a, nevertheless nontrivial, subset of the additive infinitely divisible measures. But the expression we obtain suggests that there is an inherent difference concerning the relation between the two convolutions for these domains.

Proposition 5

There exists a monomorphism of abelian semi-groups

$$\begin{aligned} {\text {EXP}}: ({\text {ID}}_c({\mathbb R},\boxplus )|_{{\mathcal {R}}_{[-1,0]}},\boxplus )\rightarrow ({\text {ID}}_c({\mathbb R}_+^*,\boxtimes ),\boxtimes ), \end{aligned}$$

given by $\mu \mapsto S^{-1}(e^{-R_{\mu }(z)})$.

Proof

For $\delta _0$ we have $R_{\delta _0}(z)\equiv 0$ which maps to $e^0=1$, corresponding to $S_{\delta _{e^0}}(z)$. Define a map ${\mathcal {R}}_{[-1,0]}\rightarrow {\mathcal {V}}$ by $R_{\mu }\mapsto -R_{\mu }$, which is injective. Then, for $v(z):=-R_{\mu }(z)$, we have

$$\begin{aligned} v({\bar{z}})=-R_{\mu }({\bar{z}})\underbrace{=}_{\hbox { by}\ 1.}-\overline{R_{\mu }(z)}=\overline{-R_{\mu }(z)}=\overline{v(z)}, \end{aligned}$$

and for $\mathfrak {I}(z)>0$ we have $\mathfrak {I}(-R(z))=-\mathfrak {I}(R(z))\le 0$. Then from [26, Theorem 3.7.3] the first part of the proposition follows. The map $\hbox {EXP}$ being a morphism of semi-groups now follows from $\exp (-(R_{\mu }+R_{\nu }))=\exp (-R_{\mu \boxplus \nu })=\exp (-R_{\mu })\cdot \exp (-R_{\nu })$. $\square$

2.2 The Witt semi-ring structure

In order to establish in the category of probability measures an analogous structure to the ring of Witt vectors, which additionally respects positivity, we have to adapt first some of the characterisations of infinitely divisible probability measures in terms of cumulants, as described in [21, Lecture 13].

So let us recall, that a complex sequence $(s_1,s_2,s_3,\dots )$ is called conditionally positive definite if the shifted sequence $(s_2,s_3,\dots )$ is positive definite, cf. [21, Notation 13.10]. If $(s_0,s_1,s_2,\dots )$ is a positive definite sequence then the truncated sequence $(s_1,s_2,s_3,\dots )$ is conditionally positive definite. Namely, for all $n\in {\mathbb N}$, and all $a_0,\dots , a_n\in {\mathbb C}$, we have, by assumption, that

$$\begin{aligned} \sum _{i,j=0}^n a_i{\bar{a}}_j s_{i+j}\ge 0. \end{aligned}$$

Let $a_0\equiv 0$. Then for $n\ge 1$ and for all $a_1,\dots ,a_n\in {\mathbb C}$, we have

$$\begin{aligned} 0\le \sum _{i,j=0}^n a_i{\bar{a}}_j s_{i+j}=\sum _{i,j=1}^n a_i{\bar{a}}_j s_{i+j}, \end{aligned}$$

(3)

as we are restricting to the sub-matrix where one deletes the first row and first column. This is a particular case of the general fact that any principal sub-matrix of a positive definite matrix is again positive definite.

Proposition 6

Let $({s_n})_{n\in {\mathbb N}^*}$, $s_n\in {\mathbb R}$, be a sequence of real numbers. Then the following statements are equivalent.

• (1) The sequence $({s_n})_{n\in {\mathbb N}^*}$ is conditionally positive definite and exponentially bounded, i.e. there exists a $C>0$ such that $|s_n|\le C^n$ for all $n\in {\mathbb N}^*$.

• (2) There exists a $\mu \in \hbox {ID}_c({\mathbb R},\boxplus )$ such that $\kappa _n(\mu )=s_n$, $n\in {\mathbb N}^*$, i.e. the real sequence is given by the free cumulants of a freely, $\boxplus$-infinitely divisible and compactly supported probability measure.

• (3) There exists a unique pair $(\gamma ,\rho )$ with $\gamma \in {\mathbb R}$ and $\rho \in \hbox {Meas}_c({\mathbb R})$ such that

  $$\begin{aligned} \sum _{n=0}^{\infty } s_{n+1}z^n=\gamma +\int _{{\mathbb R}}\frac{z}{1-xz}d\rho (x). \end{aligned}$$

Proof

$2.\Rightarrow 1.:$ This follows from [21, Theorem 13.6 and Proposition 13.15].

$1.\Rightarrow 3.:$ This follows from [21, Proposition 13.14].

$3.\Rightarrow 1.:$ This follows from [21, Lemma 13.13 and Proposition 13.14].

$1.\Rightarrow 2.:$ By [21, Proposition 13.14], there exists a finite, compactly supported, Borel measure $\rho$ on ${\mathbb R}$ such that

$$\begin{aligned} R(z):=\sum _{n=0}^{\infty } s_{n+1}z^n=s_1+\int _{{\mathbb R}}\frac{z}{1-xz}d\rho (x). \end{aligned}$$

(4)

We show that R(z) in (4) satisfies the requirements of [26, Theorem 3.7.2. (2)].

• By using a version of the “Differentiation Lemma”, cf. e.g. [16, Theorem 6.28], it follows that R(z) is analytic in $\mathbb H$ and $-\mathbb H$ and in a neighbourhood of 0. The statement for the first two regions follows directly from the integral representation.

  As the support of $\rho$ is bounded, there exists an $A>0$ such that ${\text {supp}}(\rho )\subset [-A,A]$. Let $|z|<C_{\rho }:=\min (1,1/A)$. Then $|zx|<1$ for all $x\in {\text {supp}}(\rho )$, and hence R(z) is analytic in a disc of radius $C_{\rho }$ centred at the origin.

• Positive imaginary part. We split the integral into real and imaginary parts. For $z:=a+ib$, we have

  $$\begin{aligned} \frac{z}{1-xz}=\underbrace{\frac{a-x(a^2+b^2)}{(1-ax)^2+x^2b^2}}_{=:u(x)}+i\underbrace{\frac{b}{(1-ax)^2+x^2b^2}}_{=:v(x)}=u(x)+iv(x). \end{aligned}$$

  Then

  $$\begin{aligned} R({\overline{z}})=s_1+\int _{{\mathbb R}} u(x)d\rho (x)-i\int _{{\mathbb R}}v(x)d\rho (x)=\overline{R(z)}. \end{aligned}$$

• For all $x\in {\mathbb R}$ and $b>0$, we have $v(x)>0$, and so for $z\in \mathbb H$,

  $$\begin{aligned} \mathfrak {I}(R(z))=\int _{{\mathbb R}}v(x)d\rho (x)\ge 0. \end{aligned}$$

Hence, there exists a $\mu \in \hbox {ID}_c({\mathbb R},\boxplus )$ with $R_{\mu }(z)=R(z)$. $\square$

Lemma 7

(Shifted cumulants) Let $\mu \in \hbox {ID}_c({\mathbb R},\boxplus )$ and $\kappa ({\mu })=(\kappa _n(\mu ))_{n\in {\mathbb N}^*}$ its free cumulant sequence. Then there exists a $\nu \in \hbox {ID}_c({\mathbb R},\boxplus )$ such that $\kappa _n({\nu })=\kappa _{n+2}({\mu }),$ for all $n\in {\mathbb N}^*$.

Proof

Define a sequence $(s_n)_{n\in {\mathbb N}^*}$ with $s_n:=\kappa ({\mu })_{n+2}$ for $n\in {\mathbb N}^*$ which is conditionally positive definite by (3) and exponentially bounded. Then the claim follows from Proposition 6. $\square$

For two power series, the point-wise or Hadamard multiplication, $\cdot _H$, is defined as:

$$\begin{aligned} \left( \sum _{n=0}^{\infty } a_n z^n\right) \cdot _H \left( \sum _{n=0}^{\infty } b_n z^n\right) :=\sum _{n=0}^{\infty } a_n b_b z^n~. \end{aligned}$$

The R-transform is a monomorphism $R:M({\mathbb R})\rightarrow {\mathbb R}[[z]]$ which is given by $\mu \mapsto R_{\mu }(z)$. For $\mu ,\nu \in M_c({\mathbb R})$, we define a binary operation $\boxdot$, as follows:

$$\begin{aligned} \mu \boxdot \nu :=R^{-1}(R_{\mu }(z)\cdot _H R_{\nu }(z)), \end{aligned}$$

(5)

where $R^{-1}$ is the inverse of the R-transform. Now, the following statement establishes the analytic counterpart to the purely algebraic statement in [9]. As we noted in the Introduction, the constraint given by positivity will restrict the full structure, i.e. we shall have

$$\begin{aligned} \text {ring }\rightarrow \text { semi-ring}. \end{aligned}$$

Here, cf. e.g. [6], a semi-ring is a set $(S,+,\cdot ,1,0)$ equipped with two binary operations $+$ and $\cdot$, called addition and multiplication, such that: $(S, +)$ is a commutative monoid with identity element 0, $(S,\cdot )$ is a monoid with identity element 1 and multiplication bilaterally distributes over addition. Further, one requires that $0\cdot s=s\cdot 0=0$, for all $s\in S$.

Theorem 8

$(\hbox {ID}_c({\mathbb R},\boxplus ),\boxplus ,\boxdot ,\delta _0, \nu _{\infty ,1,1})$ is a commutative semi-ring with the multiplicative unit given by the free Poisson distribution $\nu _{\infty ,1,1}$.

Proof

We have $\nu _{\infty ,1,1}\in \hbox {ID}_c({\mathbb R},\boxplus )$ which follows from, e.g. the characterisation in [26, Theorem 3.7.3, (2)], and further $R_{\nu _{\infty ,1,1}}=(1,1,1,\dots )$, which is clearly the unit for $\boxdot$.

We have to show that this product is well defined.

• (1) Let $\mu ,\nu \in \hbox {ID}_c({\mathbb R},\boxplus )$ and consider the corresponding free cumulant sequences $(k_n(\mu ))_{n\in {\mathbb N}}$ and $(k_n(\nu ))_{n\in {\mathbb N}}$. Both sequences are conditionally positive definite and by [21, Theorem 13.16, properties (1) and (2)], [21, Theorem 13.16, property (3)] and [21, Lemma 13.13] also exponentially bounded, i.e. $|\kappa _n(\bullet )|\le C_{\bullet }^n$, $n\in {\mathbb N}$ for some constant $C_{\bullet }>0$, depending on the measures $\mu$ and $\nu$, respectively.

  Hence, $|\kappa _n(\mu )\cdot \kappa _n(\nu )|\le (C_{\mu }\cdot C_{\nu })^n$ for all $n\in {\mathbb N}$, i.e. the point-wise product of the free cumulants does not grow faster than exponentially.

• (2) In order to show positivity, we consider the corresponding Hankel matrices. Let ${\tilde{\kappa }}_n(\bullet ):=\kappa _{n+2}(\bullet )$ for $n\in {\mathbb N}$. Then, by the Schur Product Theorem,

  $$\begin{aligned} ({\tilde{\kappa }}_{i+j}(\mu )\cdot {\tilde{\kappa }}_{i+j}(\nu ))_{0\le i,j\le n}, \end{aligned}$$

  is positive definite.

  Therefore the sequence $(h_n)_{n\in {\mathbb N}^*}$, with $h_1:=\kappa _1(\mu )\cdot \kappa _1(\nu )$ and $h_n:=\kappa _n(\mu )\cdot \kappa _n(\nu )$, $n\in {\mathbb N}^*$, is conditionally positive definite and exponentially bounded.

  By Proposition 6 there exists a probability measure $\xi \in \hbox {ID}_c({\mathbb R},\boxplus )$ such that $h_n=\kappa _n(\xi )$,$n\in {\mathbb N}^*$. Finally, let $\mu \boxdot \nu :=\xi$.

• (3) Associativity and distributivity follow from the fact that the operations are given component-wise, which individually satisfy these properties. Apply R to $(\mu \boxplus \nu )\boxdot \eta$, which by definition, yields, $R_{\mu \boxplus \nu }\cdot _H R_{\eta }=(R_{\mu }+R_{\nu })\cdot _H R_{\eta }$ from which the result $\mu \boxdot \eta \boxplus \nu \boxdot \eta$, after applying $R^{-1}$, follows.

$\square$

We remark that the binary operation $\boxdot$ can be lifted via the exponential map to $\hbox {ID}^*(S^1,\boxtimes )$ and $\hbox {ID}_c({\mathbb R}_+^*,\boxtimes )$, respectively. However, in the analytic case some additional considerations are necessary.

Proposition 9

The family of Dirac distributions is a two-sided ideal in $\hbox {ID}_c({\mathbb R},\boxplus )$. We have

$$\begin{aligned} \delta _a\boxplus \delta _b= & {} \delta _{a+b},\\ \delta _a\boxdot \delta _b= & {} \delta _{ab}. \end{aligned}$$

Further for $\mu \in \hbox {ID}({\mathbb R},\boxplus )$ we have

$$\begin{aligned} \delta _a\boxdot \mu =\delta _{\kappa _1(\mu )\cdot a} \end{aligned}$$

where $\kappa _1(\mu )$ is the first free cumulant of $\mu$.

For $a,b\in {\mathbb R}$ and $r,s>0$, the semicircle law centred at a and of radius r, is defined as the distribution $\gamma _{a,r}:{\mathbb C}[z]\rightarrow {\mathbb C}$, given by

$$\begin{aligned} \gamma _{a,r}(z^n):=\frac{2}{\pi r^2}\int _{a-r}^{a+r} t^n\sqrt{r^2-(t-a)^2}\,dt\qquad \forall n\in {\mathbb N}, \end{aligned}$$

and its R-transform is

$$\begin{aligned} {R}_{\gamma _{a,r}}(z)=a+\frac{r^2}{4}z~. \end{aligned}$$

(6)

Proposition 10

Let $\gamma _{a,0}:=\delta _a$. The family of semicircular distributions is a two-sided ideal in $\hbox {ID}_c({\mathbb R},\boxplus )$, which contains the Dirac distributions as sub-ideal. We have

$$\begin{aligned} \gamma _{a,r}\boxplus \gamma _{b,s}= & {} \gamma _{a+b,\sqrt{r^2+s^2}}, \nonumber \\ \gamma _{a,r}\boxdot \gamma _{b,s}= & {} \gamma _{ab, rs/2}. \end{aligned}$$

(7)

Proof

We have ${R}_{\gamma _{a,r}\boxdot \gamma _{b,s}}(z)=abz+\frac{r^2 s^2}{4\cdot 4}z^2$ by component-wise multiplication of the R-transforms and the relation (6).

For any $\mu \in M_c({\mathbb R})$, $\kappa (\mu )=(\kappa _1(\mu ),\kappa _2(\mu ),\dots )$ with $\kappa _1(\mu )\in {\mathbb R}$ and $\kappa _2(\mu )\ge 0$ as, e.g. by [ [21] Proposition 13.14], $\kappa _2(\mu )=m_0(\rho )=\rho ({\mathbb R})\ge 0$ for some compactly supported finite measure $\rho$. Therefore, the non-negativity of the second free cumulant is always preserved under the commutative point-wise multiplication, and therefore the property of being an ideal follows. $\square$

Proposition 11

The family of free Poisson distributions $\mathbf {fP}$ with rate $\lambda \in {\mathbb R}_+$ and jump size $\alpha \in {\mathbb R}$ is closed with respect to the $\boxdot$-convolution, i.e. for $\lambda ,\lambda '\in {\mathbb R}_+$ and $\alpha ,\beta \in {\mathbb R}$ we have

$$\begin{aligned} \nu _{\infty ,\lambda ,\alpha }\boxdot \nu _{\infty ,\lambda ',\beta }=\nu _{\infty ,\lambda \lambda ',\alpha \beta }. \end{aligned}$$

Therefore, $\mathbf {fP}$ is a $\boxdot$-monoid, and for $(\lambda ,\alpha )\in {\mathbb R}^*_+\times {\mathbb R}^*$, an abelian group.

Proof

The statement follows from the representation given in Eq. (2) by component-wise multiplication. $\square$

Let us remark that a more general statement should be valid which holds also for the compound free Poisson distribution, cf. [21, p. 206].

We derived our analytic results for compactly supported probability measures, but it is natural to assume that one can extend them to all freely infinitely divisible probability measures.

2.3 Endomorphisms

As we have now established the Witt semi-ring structure for freely infinitely divisible probability measures, we shall investigate elements of its ring of endomorphisms in analogy with the theory of Witt vectors, cf. e.g. [14].

For $a\in {\mathbb R}$, we define the Teichmüller representative $\tau$ as:

$$\begin{aligned} \tau :{\mathbb R}\rightarrow {\text {ID}}_c({\mathbb R},\boxplus ),\quad a\mapsto \nu _{\infty ,1,a}, \end{aligned}$$

which gives a group isomorphism

$$\begin{aligned} \tau :({\mathbb R}^*,\cdot )\rightarrow (\nu _{\infty ,1,\bullet },\boxdot ). \end{aligned}$$

In general, the multiplicative map $A\rightarrow {\mathbb {W}}(A):= 1+zA[[z]]$ such that $a\mapsto [a]:= (1-az)$, is called the Teichmüller character.

Let us define the squared shift or décalage operator ${\hat{\partial }}^2$. For $\mu \in \hbox {ID}_c({\mathbb R},\boxplus )$, with free cumulants $\kappa ({\mu })=(\kappa _n({\mu }))_{n\in {\mathbb N}^*}$, let ${\hat{\partial }}^2(\mu )\in \hbox {ID}_c({\mathbb R},\boxplus )$, such that for all $n\in {\mathbb N}^*$, we have:

$$\begin{aligned} \kappa _n(\partial ^2(\mu )):=\kappa _{n+2}(\mu ), \end{aligned}$$

which by Lemma 7 is well-defined. The notation ${\hat{\partial }}^2$ emphasises the fact that in order to have the décalage operator acting properly on measures we have to take its square, i.e. it is elliptic.

Proposition 12

The operator ${\hat{\partial }}^2$ defines a semi-ring endomorphism of $\hbox {ID}_c({\mathbb R},\boxplus )$, i.e. for $\mu _1,\mu _2\in \hbox {ID}_c({\mathbb R},\boxplus )$, we have:

$$\begin{aligned} {\hat{\partial }}^2(\mu _1\boxplus \mu _2)= & {} {\hat{\partial }}^2(\mu _1)\boxplus \partial ^2(\mu _2),\\ {\hat{\partial }}^2(\mu _1\boxdot \mu _2)= & {} {\hat{\partial }}^2(\mu _1)\boxdot {\hat{\partial }}^2(\mu _2). \end{aligned}$$

In particular, it respects the additive and multiplicative unit, i.e. we have

$$\begin{aligned} {\hat{\partial }}^2(\delta _0)= & {} \delta _0,\\ {\hat{\partial }}^2(\nu _{\infty ,1,1})= & {} \nu _{\infty ,1,1}. \end{aligned}$$

Proof

From $\kappa _n(\mu _1\boxplus \mu _2)=\kappa _n(\mu _1)+\kappa _n(\mu _2)$ and, by definition $\kappa _n(\mu _1\boxdot \mu _2)=\kappa _n(\mu _1)\cdot \kappa _n(\mu _2)$, $n\in {\mathbb N}^*$, the claim follows. $\square$

We introduce the Frobenius endomorphism as follows. For $a\in A^{{\mathbb N}^*}$, $n\in {\mathbb N}$, with $0^0:=1$, let ${\mathbf {f}}_n:A^{{\mathbb N}^*}\rightarrow A^{{\mathbb N}^*}$, and define

$$\begin{aligned} (a_j)_{j\in {\mathbb N}^*}\mapsto (a_j^n)_{j\in {\mathbb N}^*}. \end{aligned}$$

(8)

Proposition 13

For $\mu \in \hbox {ID}_c({\mathbb R},\boxplus )$, $n\in {\mathbb N}$, we define

$$\begin{aligned} {\mathbf {f}}_n(\mu ):=R^{-1}({\mathbf {f}}_n(\kappa (\mu ))). \end{aligned}$$

Then the following holds:

$$\begin{aligned} {\mathbf {f}}_n(\mu )=\mu ^{\boxdot n}, \end{aligned}$$

(9)

and ${\mathbf {f}}_n$ yields a $\boxdot$-endomorphisms, i.e. for $\mu ,\nu \in \hbox {ID}_c({\mathbb R},\boxplus )$:

$$\begin{aligned} {\mathbf {f}}_n(\mu \boxdot \nu )= & {} {\mathbf {f}}_n(\mu )\boxdot {\mathbf {f}}_n(\nu ),\\ (\mu \boxdot \nu )^{\boxdot n}= & {} \mu ^{\boxdot n}\boxdot \nu ^{\boxdot n}. \end{aligned}$$

Proof

${\mathbf {f}}_n(\kappa (\mu ))=(\kappa _j(\mu )^n)_{j\in {\mathbb N}^*}=\kappa (\mu )^{\boxdot n}$ from which the claim follows. $\square$

We remark that the set of free Poisson distributions is invariant with respect to $\partial ^2$ and ${\mathbf {f}}_n$, in particular

$$\begin{aligned} {\hat{\partial }}^2(\nu _{\infty ,\lambda ,\alpha })=\nu _{\infty ,\lambda \alpha ^2,\alpha }, \end{aligned}$$

holds.

2.4 Geometric properties

The space of infinitely divisible probability measures possesses, due to the Lévy-Khintchine representation, a remarkable geometric structure.

Proposition 14

There exists a surjection $\pi :\hbox {ID}_c({\mathbb R},\boxplus )\rightarrow \hbox {Meas}_c({\mathbb R})$, with fibre

$$\begin{aligned} \pi ^{-1}(\rho )\cong {\mathbb R}\times \{\rho \}\cong {\mathbb R}. \end{aligned}$$

The space $\hbox {ID}_c({\mathbb R},\boxplus )$ carries two actions, namely for $\mu \in \hbox {ID}_c({\mathbb R},\boxplus )$, $r\in {\mathbb R}$ and $c\in {\mathbb R}_+^*$, we have:

• the ${\mathbb R}$-action,(shift along the fibre) is given by:

  $$\begin{aligned} r.\mu :=\delta _{r}\boxplus \mu , \end{aligned}$$

• the ${\mathbb R}^*_+$-action (scaling) is given by:

  $$\begin{aligned} c.\mu :=\nu _{\infty ,c,1}\boxdot \mu . \end{aligned}$$

Proof

The projection $\pi$ is given by the representation (4). Further, we have $(r+r').\mu =\delta _{r+r'}\boxplus \mu =\delta _r\boxplus (\delta _{r'}\boxplus \mu )$ and $0.\mu =\delta _0\boxplus \mu =\mu$. $\square$

So, for the Dirac, semi-circular and free Poisson distribution, with $c,r,\lambda \ge 0$, and $\alpha \in {\mathbb R}$, we have

$$\begin{aligned} c.\delta _a= & {} \delta _{c\cdot a},\\ c.\gamma _{a,r}= & {} \gamma _{c\cdot a,{\sqrt{c}}\cdot r},\\ c.\nu _{\lambda ,\alpha }= & {} \nu _{c\cdot \lambda ,\alpha }, \end{aligned}$$

which shows that these classes of measures are invariant under the action. In particular, the semi-circular distributions form a convex sub-cone.

2.5 Relation with classically infinitely divisible probability measures

Here we shall discuss the relation of our results with classically infinitely divisible measures, which due to the Bercovici–Pata bijection [3] is possible. The references for this section are [2, 3, 7]. Let, $\hbox {ID}({\mathbb R},*)$ and $\hbox {ID}({\mathbb R},\boxplus )$ denote the set of $*$-infinitely (classically) divisible and $\boxplus$-infinitely (freely) divisible probability measure on the real line, respectively. Let us emphasise, here we do not assume any restriction on the support.

We have $\hbox {ID}_c({\mathbb R},\boxplus )\subset \hbox {ID}({\mathbb R},\boxplus )$ as a dense sub-monoid and that there is a bijection $\varphi$ between characteristic pairs $(\gamma ,\sigma )$ and $\bullet$-infinitely divisible probability measures on ${\mathbb R}$, where $\bullet =*,\boxplus$, i.e. we have

$$\begin{aligned} \varphi _{{\text {cp}}}:{\mathbb R}\times {\text {Meas}}({\mathbb R})\rightarrow {\text {ID}}({\mathbb R},\bullet ). \end{aligned}$$

(10)

Let $\mu \in \hbox {ID}({\mathbb R},\boxplus )$ with characteristic pair $(\gamma ,\sigma )$. Then from [2, Equation (4.5)] we obtain

$$\begin{aligned} R_{\mu }(z)=\gamma +\int _{{\mathbb R}}\frac{z+x}{1-xz}d\sigma (x). \end{aligned}$$

(11)

Proposition 15

Let $\pi _*:\hbox {ID}({\mathbb R},*)\rightarrow \hbox {Meas}({\mathbb R})$ and $\pi _{\boxplus }:\hbox {ID}({\mathbb R},\boxplus )\rightarrow \hbox {Meas}({\mathbb R})$ be the projections onto the second factor of the characteristic pair, i.e. $\pi _i(\mu ):={\text {pr}}_2(\gamma ,\sigma )=\sigma$, $i=*,\boxplus$ where $\mu$ is represented by $(\gamma ,\sigma )$. The fibre $\pi ^{-1}_i(\sigma )$ is isomorphic to ${\mathbb R}$, and the Bercovici–Pata bijection gives a bundle map $\varphi _{\hbox {BP}}:{\text {ID}}({\mathbb R},*)\rightarrow \hbox {ID}({\mathbb R},\boxplus )$, i.e. the following diagram

commutes.

For $\mu ,\nu \in \hbox {ID}({\mathbb R},*)$ with classical cumulants $(c(\mu ))_{n\in {\mathbb N}^*}$ and $(c(\nu ))_{n\in {\mathbb N}^*}$, we define, equivalently to (5), a binary operation $\star$ as follows:

$$\begin{aligned} c(\mu \star \nu )_n:=c(\mu )_n\cdot c(\nu )_n, \end{aligned}$$

(12)

i.e. the two cumulant sequences are multiplied component-wise, and then the measure is obtained by the inverse Fourier transform of the exponential of the resulting sequence.

We have:

Distribution	Parameters	Classical cumulants
Dirac	$a\in {\mathbb R}$	(a,0,0,0,...)
Normal	$m\in {\mathbb R}$, $\sigma ^2\in {\mathbb R}^*_+$	$(m,\sigma ^2,0,0,\dots )$
Poisson	$\lambda \in {\mathbb R}_+^*$	$(\lambda ,\lambda ,\lambda ,\lambda ,\dots )$

Proposition 16

The joint set of Dirac, normal and Poisson distributions form a commutative monoid for the operation $\star$, with unit the Poisson distribution with parameter $\lambda =1$.

Let us note that in equality (4), the pair $(s_1,\rho )$ also uniquely characterises $\mu \in \hbox {ID}_c({\mathbb R},\boxplus )$. From expression (11) we obtain the relation $\kappa _1(\mu )=\gamma +\int _{{\mathbb R}}xd\sigma (x)$, and for the measures $\sigma$ and $\rho$ we have in general

$$\begin{aligned} \int _{{\mathbb R}}\frac{z+x}{1-xz}d\sigma (x)\ne \int _{{\mathbb R}}\frac{z}{1-xz}d\rho (x). \end{aligned}$$

Proposition 17

The following statements hold:

• (1) The sets $\hbox {ID}({\mathbb R},*)$ and $\hbox {ID}({\mathbb R},\boxplus )$ are convex cones for the following operation: for $\alpha ,\beta \ge 0$ and $\mu ,\nu \in \hbox {ID}({\mathbb R},\bullet )$ with $\bullet =*,\boxplus$, let

  $$\begin{aligned} \alpha \mu +\beta \nu :=\varphi ^{-1}(\alpha \gamma _{\mu }+\beta \gamma _{\nu },\alpha \mu +\beta \nu ) \end{aligned}$$

  where $\varphi _{{\text {cp}}}^{-1}$ is the inverse of the bijection (10).

• (2) The normal and the semicircular distributions form isomorphic convex sub-cones of ${\text {ID}}({\mathbb R},\bullet )$.

• (3) ${\text {ID}}({\mathbb R},\bullet )$, $\bullet =*,\boxplus$, carries a fibre-wise ${\mathbb R}$-action, which is given by

  $$\begin{aligned} \delta _a.\mu :=\varphi _{{\text {cp}}}^{-1}(\gamma +a,\rho ), \end{aligned}$$

  where $a\in {\mathbb R}$ and $\varphi _{{\text {cp}}}^{-1}$ is the compositional inverse of the bijection (10).

2.6 Some relations with limit theorems

Here we shall briefly discuss the link operations on $M_c({\mathbb R})$, via moments, and operations on $\hbox {ID}_c({\mathbb R},\boxplus )$, via free cumulants, have as infinitesimal limits, given by the relation

$$\begin{aligned} \lim _{N\rightarrow \infty } \mu _N^{\boxplus N}=\mu . \end{aligned}$$

Let us summarise [21, Remark 13.4] and [21, Theorem 13.16 (4)] as

Proposition 18

Let $\mu ,\mu _N\in M_c({\mathbb R})$, $N\in {\mathbb N}^*$, be given. Then

$$\begin{aligned} \mu _N^{\boxplus N}\xrightarrow [N\rightarrow \infty ]{\text {law}}\mu\Leftrightarrow & {} \lim _{N\rightarrow \infty } N\cdot m_n(\mu _N)=\kappa _n(\mu ),\quad \forall n\in {\mathbb N}^*, \end{aligned}$$

where $m_n()$ is the nth moment, and in this case $\mu \in {\text {ID}}_c({\mathbb R},\boxplus )$, i.e. $\mu$ is $\boxplus$-infinitely divisible.

Let ${\text {mult}}:{\mathbb R}^2\rightarrow {\mathbb R}$, $(x,y)\mapsto x\cdot y$ be the multiplication map. For $\mu ,\nu \in M_c({\mathbb R})$ define

$$\begin{aligned} \mu \cdot \nu :={\text {mult}}_*(\mu \otimes \nu ), \end{aligned}$$

which is the image measure on ${\mathbb R}$ of the tensor product measure $\mu \otimes \nu$ on ${\mathbb R}^2$, under multiplication of the coordinates, i.e.

$$\begin{aligned} \mu \cdot \nu (A)=\int _{{\mathbb R}}\int _{{\mathbb R}}\chi _A(xy)d\mu (x) d\nu (y),\quad A\in {\mathcal {B}}({\mathbb R}^2). \end{aligned}$$

Then $\mu \cdot \nu \in M_c({\mathbb R})$, and further, for all $n\in {\mathbb N}$,

$$\begin{aligned} m_n(\mu \cdot \nu )=m_n(\mu )\cdot m_n(\nu ), \end{aligned}$$

holds.

Let, $\lim _{N\rightarrow \infty }\mu _N^{\boxtimes N}=\mu$, and $\lim _{N\rightarrow \infty }\nu _N^{\boxtimes N}=\nu$, be given as in Proposition 18. Then

$$\begin{aligned} \lim _{N\rightarrow \infty } N^2\cdot m_n(\mu _N\cdot \nu _N)= & {} \left( \lim _{N\rightarrow \infty }N\cdot m_n(\mu _N)\right) \cdot \left( \lim _{N\rightarrow \infty } N\cdot m_n(\nu _N)\right) \\= & {} \kappa _n(\mu )\cdot \kappa _n(\nu )\\= & {} \kappa _n(\mu \boxdot \nu ). \end{aligned}$$

Hence for $N\rightarrow \infty$, we have:

$$\begin{aligned} \left( \mu _N\cdot \nu _N\right) ^{\boxplus N^2}\xrightarrow {law}\mu \boxdot \nu . \end{aligned}$$

Similarly, for $q\in [0,1]$, we have

$$\begin{aligned} N\cdot m_n\left( q\mu _N+(1-q)\nu _N\right) =q N\cdot m_n(\mu _N)+(1-q)N\cdot m_n(\nu ), \end{aligned}$$

and so, for $N\rightarrow \infty$, we get

$$\begin{aligned} q\kappa _n(\mu )+(1-q)\kappa _n(\nu )=:\kappa _n(\mu +_q\nu ), \end{aligned}$$

where the binary operation $+_q$ shall be (properly) introduced in  3.1. Hence,

$$\begin{aligned} \left( q\mu _N+(1-q)\nu _N\right) ^{\boxplus N}\xrightarrow [N\rightarrow \infty ]{}\mu +_q\nu . \end{aligned}$$

Finally, define

$$\begin{aligned} d\xi _N:=\frac{1}{m_2(\mu _N)}x^2d\mu _N, \end{aligned}$$

where $m_2(\mu _N)$ is the variance of $\mu _N$. Then

$$\begin{aligned} m_n(\xi _N)=\frac{N}{N}\cdot \frac{m_{n+2}(\mu _N)}{ m_2(\mu _N)}\xrightarrow [N\rightarrow \infty ]{}\frac{\kappa _{n+2}(\mu )}{\kappa _2(\mu )}=\frac{\kappa _n({\hat{\partial }}^2(\mu ))}{\kappa _2(\mu )}. \end{aligned}$$

Hence

$$\begin{aligned} \lim _{N\rightarrow \infty }\xi _N^{\boxplus N/N}=\left( {\hat{\partial }}^2(\mu )\right) ^{\boxplus \frac{1}{\kappa _2(\mu )}}. \end{aligned}$$

Let us summarise these correspondences in the following table:

$M_c({\mathbb R})$	${\text {ID}}_c({\mathbb R},\boxplus )$
$\mu \cdot \nu$	$\mu \boxdot \nu$
$q\mu +(1-q)\nu$	$\mu +_q\nu$
$x^2 d\mu (x)/\int _{{\mathbb R}}x^2d\mu (x)$	$\left( {\hat{\partial }}^2(\mu )\right) ^{\boxplus \frac{1}{\kappa _2(\mu )}}$

Now, by forming algebraic expressions, we obtain further operations which transfer into infinitesimal limits.

3 Algebraic theories related to probability measures

In the previous sections our investigation has been strongly motivated by the relation free probability has with the ring of Witt vectors. Here we shall consider a more general framework, namely the (co)-monadic approach introduced in [12].

3.1 Convex spaces

Let us discuss the relations the barycentric calculus has with probability measures, in particular infinitely divisible ones. First, we state the following

Proposition 19

For $\mu ,\nu \in {\text {Meas}}({\mathbb R})$ such that all moments exists and $q\in [0,1]$, let $\lambda :=q\mu +(1-q)\nu \in {\text {Meas}}({\mathbb R})$ be the q-convex linear combination of the measures $\mu$ and $\nu$. Then the moment map m is a morphism for convex linear combinations, i.e. the moments $(m_n(\lambda ))_{n\in {\mathbb N}}$ satisfy:

$$\begin{aligned} m_n(q\mu +(1-q)\nu )=qm_n(\mu )+(1-q) m_n(\nu ). \end{aligned}$$

Proof

This follows from

$$\begin{aligned} \int _{{\mathbb R}} x^n d(q\mu +(1-q)\nu )=q\int _{{\mathbb R}}x^nd\mu +(1-q)\int _{{\mathbb R}}x^nd\nu . \end{aligned}$$

$\square$

Let us briefly recall the abstract notion of a convex space, cf. [13, 15].

Definition 5

Let C be as set. The structure of a convex space on C is given by, for every $q\in [0,1]$ a binary operation $+_q$, the barycentric addition, which for $x,y,z\in C$ satisfies the following rules:

• (1) $+_q(x,y)=+_{1-q}(y,x),$

• (2) $+_q(x,x)=x,$

• (3) $+_0(x,y)=y,$

• (4) $+_p(+_q(x,y),z)=+_{p+(1-p)q}(+_{\frac{p}{p+(1-p)q}}(x,y),z)$,   if $p+(1-p)q\ne 0$.

For $\mu \in {\text {ID}}_c({\mathbb R},\boxplus )$ and $t\in {\mathbb R}^*_+$, let $\mu ^{\boxplus t}$ denote the convolution semigroup in the space of compactly supported probability measures on ${\mathbb R}$, which is the continuum limit of the rational convolution powers $\mu ^{\boxplus m/n}$, $m,n\in {\mathbb N}^*$, as explained in [21, Remark 13.7. 2)]. We set $\mu ^{\boxplus 0}:=\delta _0$.

Proposition 20

For $\mu ,\nu \in {\text {ID}}_c({\mathbb R},\boxplus )$ and $\alpha ,\beta \in {\mathbb R}_+$, define the family of binary operations $+_{\alpha ,\beta }$ as follows:

$$\begin{aligned} \mu +_{\alpha ,\beta }\nu :=R^{-1}(\alpha R_{\mu }+\beta R_{\nu }), \end{aligned}$$

which implies

$$\begin{aligned} \mu +_{\alpha ,\beta }\nu =\mu ^{\boxplus \alpha }\boxplus \nu ^{\boxplus \beta }. \end{aligned}$$

Therefore, $({\text {ID}}_c({\mathbb R},\boxplus ),+_{\alpha ,\beta })$ has the structure of a convex cone, and for $\alpha +\beta =1$ of a convex space.

Proof

This follows from Lemma 1. The second proof follows from the equality (11). $\square$

Proposition 21

For $q\in [0,1]$, let $+_q:=+_{q,(1-q)}$. The operations $\boxplus$ and $\boxdot$ distribute over the barycentric addition $+_q$, i.e. for $\mu ,\nu ,\xi \in {\text {ID}}_c({\mathbb R},\boxplus )$, we have:

$$\begin{aligned} (\mu +_q\nu )\boxplus \xi =(\mu \boxplus \xi )+_q(\nu \boxplus \xi ),\\ (\mu +_q\nu )\boxdot \xi =(\mu \boxdot \xi )+_q(\nu \boxdot \xi ).\\ \end{aligned}$$

Proof

In order to show distributivity, we calculate:

$$\begin{aligned} R\left( (\mu +_q\nu )\boxplus \xi \right) =(qR_{\mu }+(1-q)R_{\nu })+R_{\xi }=q(R_{\mu }+R_{\xi })+(1-q)(R_{\nu }+R_{\xi }), \end{aligned}$$

and

$$\begin{aligned} R\left( (\mu +_q\nu )\boxdot \xi \right) =(qR_{\mu }+(1-q)R_{\nu })\cdot _HR_{\xi } = q(R_{\mu }\cdot _HR_{\xi })+(1-q)(R_{\nu }\cdot _HR_{\xi }). \end{aligned}$$

$\square$

Let us point out that we have to distinguish between operations defined on moments and cumulants and in general we can not mix the two, as the following example shows, cf. [21, Remarks 12.10 (2)]. For $\mu :=\frac{1}{2}(\delta _{-1}+\delta _{+1})$, with the convex combination taken for moments, we have

$$\begin{aligned} \frac{1}{2}(\delta _{-1}+\delta _{+1})\boxplus \mu \ne \frac{1}{2}(\delta _{-1}\boxplus \mu )+\frac{1}{2}(\delta _{+1}\boxplus \mu ). \end{aligned}$$

3.2 The finite Giry monad

Let S be a semi-ring and $G_S$ be the functor

$$\begin{aligned} G_S:\mathbf {Set}\rightarrow \mathbf {Set} \end{aligned}$$

such that for $X\in \mathbf {Set}$, we have

$$\begin{aligned} G_S(X):=\left\{ \varphi :X\rightarrow S|~\varphi (x)=0\text { for almost all }x\in X,\text { and}\sum _{x\in X}\varphi (x)=1\right\} . \end{aligned}$$

Then every $\varphi \in G_S(X)$ has a representation

$$\begin{aligned} \varphi =s_1x_1+\cdots +s_nx_n, \end{aligned}$$

with $s_i\in S$, $x_i\in X$, $i=1,\dots ,n$, and where $s_i=0$ is possible and the $x_i$ are not necessarily different, i.e. we allow also non-reduced formal sums, which can be considered as special S-valued divisors.

Proposition 22

[13, 15] Let S be a commutative unital semiring. The distribution or finite Giry monad $(G_S,\eta ,\mu )$ with coefficients in S, is given by:

$$\begin{aligned}&\eta _X:X\rightarrow G_S(X),\\&x \mapsto 1_Sx,\\&\mu _X : G_S^2:=G_S\circ G_S\rightarrow G_S,\\&\mu _X\left( \sum _{i=1}^n s_i\varphi _i\right) := x\mapsto \sum _{i=1}^n s_i\varphi _i(x) \end{aligned}$$

The corresponding statement for $\boxplus$-infinitely divisible measures is as follows.

Proposition 23

$({\text {ID}}_c({\mathbb R},\boxplus ),+_q, q\in [0,1])$ is an algebra for the finite Giry / distribution monad, with $\alpha :G_{{\mathbb R}_+}({\text {ID}}_c({\mathbb R},\boxplus ))\rightarrow {\text {ID}}_c({\mathbb R},\boxplus )$ given by:

$$\begin{aligned} \alpha (\lambda _1\mu _1+\cdots \lambda _n x_n):=\mu _1+_{q_1}(\mu _2+_{\mu _2}(\cdots +_{q_{n-2}}(\mu _{n-1}+_{q_{n-1}}\mu _n))) \end{aligned}$$

for $\mu _i\in {\text {ID}}_c({\mathbb R},\boxplus )$, $\lambda _i\in {\mathbb R}_+$, $\sum _{i=1}^n\lambda _i=1$, $q_1:=\lambda _1$ and $q_i:=\frac{q_i}{1-q_1-\dots -q_{i-1}}$, $i=1,\dots , n$.

Proof

The statement follows from [15, Theorem 4]. $\square$

3.3 The convolution algebra

We shall consider, following the notation in [18], an algebraic theory $(\Omega ,E)$ consisting of a graded set $\Omega$ of operations and E a set of identities which together define an $\langle \Omega ,E\rangle$-algebra. Surprisingly, this algebra , which should abstractly capture the operations we have encountered, is equally related to a number of other algebraic structures, such as differential algebras, endomorphism algebras and Witt vectors, cf. [12].

Definition 6

The (partial) convolution algebra is given by the following generators $\Omega$ and relations E:

• $\Omega (0):=\{0,1\}\times {\mathbb R}$, nullary operations,

• $\Omega (1):=\{{\text {id}}, \partial , f_n\}$, $n\in {\mathbb N}$, unary operations,

• $\Omega (2):=\{+,\cdot ,+_q\}$, $q\in \Delta ^1$, where $\Delta ^1$ denotes the geometric one-simplex, binary operations.

These satisfy the following relations E:

• (1) $(+,\cdot ,{\mathbf {0}}:=(0,0),{\mathbf {1}}:=(1,1))$ forms a commutative semi-ring with additive unit ${\mathbf {0}}$ and multiplicative unit ${\mathbf {1}}$,

• (2) $+_q$$q\in [0,1]$ satisfies the properties of Definition 5,

• (3) ${\partial }$ is an endomorphism for $+,\cdot$ and $+_q$, $q\in [0,1]$,

• (4) $f_n$$n\in {\mathbb N}^*$ is an endomorphism for $\cdot$.

We call this structure “partial” because there exist further classes of operations, however here we capture those which we have met and which have been relevant within the present context. Others could be obtained from the moment-cumulant duality.

Also, the above algebraic structure is more general than an operad, as its definition involves co-operations. However, it contains several interesting operads, e.g. Definition 5, without property 2. defines an operad called the simplex operad, as the derived n-ary operations contain the standard $(n-1)$-simplex as label, cf. the essay by T. Leinster [17]. This can be described as follows:

The set of operations we found fits into the algebraic structure which we summarise next. Let us, define the operations $+:=\boxplus$, $\cdot :=\boxdot$, $\partial ^2:={\hat{\partial }}^2$, ${\mathbf {f}}_n$ as in (8) and $+_q$ as in Proposition 21. Then we have

Corollary 24

With the operations defined above, the following holds:

• (1) $({\text {ID}}_c({\mathbb R},\boxplus ), \boxplus ,\boxdot ,\delta _0,\nu _{\infty ,1,1}, {\hat{\partial }}^2,{\mathbf {f}}_n, n\in {\mathbb N}^*)$, has the structure of an algebra over the set operad generated by $\Omega (0),\Omega (1)$ and $+,\cdot$.

• (2) $({\text {ID}}_c({\mathbb R},\boxplus ), \boxplus ,\boxdot ,+_q, q\in [0,1], {\hat{\partial }}^2,f_n,n\in {\mathbb N})$ is an $(\Omega ,E)$-algebra, with the operations and relations specified in Definition 6.