Moments and cumulants in infinite dimensionswith applications toPoisson, Gamma and Dirichlet–Fergusonrandom measures\thanksrefT1

# Moments and cumulants in infinite dimensions with applications to Poisson, Gamma and Dirichlet–Ferguson random measures\thanksrefT1

\fnmsLorenzo \snmDello Schiavo\thanksreft1label=e1]delloschiavo@iam.uni-bonn.de [ University of Bonn Institut für Angewandte Mathematik
Rheinische Friedrich-Wilhelms-Universität Bonn
Endenicher Allee 60
DE 53115 Bonn
Germany
###### Abstract

We show that the chaos representation of some Compound Poisson Type processes displays an underlying intrinsic combinatorial structure, partly independent of the chosen process. From the computational viewpoint, we solve the arising combinatorial complexity by means of the moments/cumulants duality for the laws of the corresponding processes, themselves measures on distributional spaces, and provide a combinatorial interpretation of the associated ‘extended’ Fock spaces. From the theoretical viewpoint, in the case of the Gamma measure, we trace back such complexity to its ‘simplicial part’, i.e. the Dirichlet–Ferguson measure, hence to the Dirichlet distribution on the finite-dimensional simplex. We thoroughly explore the combinatorial and algebraic properties of the latter distribution, arising in connection with cycle index polynomials of symmetric groups and dynamical symmetry algebras of confluent Lauricella functions.

[
\kwd
\setattribute

journalname \arxiv \startlocaldefs \endlocaldefs

\runtitle

Moments & cumulants in infinite dimensions

{aug}\thankstext

T1This work has been jointly supported by the CRC 1060 The Mathematics of Emergent Effects at the University of Bonn, funded through the Deutsche Forschungsgemeinschaft, and by the Hausdorff Center for Mathematics at the University of Bonn, funded through the German Excellence Initiative.

\thankstext

t1I gratefully acknowledge fruitful conversations with P. Ferrari; K.-T. Sturm and M. Huesmann; E. Lytvynov, M. Gubinelli and L. Borasi; S. Albeverio, M. Gordina and C. Stroppel, on some aspects of this work respectively concerning combinatorics, probability theory, infinite-dimensional analysis and representation theory. I am also indebted to A. Nota for having introduced me to the concept of cumulants, to E. Lytvynov and W. Miller for pointing out to me the references [35] and [50] respectively, and finally to M. Huesmann for partly proofreading a preliminary version of this work.

July 17, 2019

class=MSC] \kwd[Primary ]60E05 \kwd[; secondary ]60G57 \kwd62E10 \kwd46G12 \kwd33C65 \kwd33C67 \kwd30H20 \kwd05A17 \kwd60H05.

moments \kwdcumulants \kwdcompound Poisson processes \kwdWiener–Itô chaos \kwdDirichlet distribution \kwdDirichlet–Ferguson process \kwdGamma measure \kwdextended Fock space \kwdhypergeometric Lauricella functions \kwdcycle index polynomials of symmetric groups \kwddynamical symmetry algebras of hypergeometric functions.

## 1 Introduction

Given a probability distribution  on a linear space , denote by

 L[ν](tX)\coloneqq ∫Φexp[tX(x)]dν(x), resp. by K[ν](tX)\coloneqq lnL[ν](tX),

its moment, resp. cumulant, generating function computed at a real-valued random variable . The coefficients in the corresponding McLaurin expansions in , respectively the (raw) moments  and the cumulants , represent a cornerstone in elementary probability, with both broad-ranging applications and great interest per se (e.g. – for the cumulants – in free probability [5, 39, 40, and ref.s therein] and kinetic derivation theory [45, 44]; also, see [27] for historical remarks). Among the inherent problems, that of obtaining moments from cumulants is seemingly a simple one, usually reduced to the observation that

 μ′n=Bn[κ1,…,κn],

where denotes the -variate (complete) Bell polynomial. Similar formulae hold for obtaining cumulants from moments, raw moments from central moments and vice versa, thus suggesting for these correspondences the – somewhat informal – designation of dualities.

• appropriate generalizations of such dualities (Thm. 3.3) provide a better understanding (e.g. Rem.s 5.8&6.3) and simpler proofs (e.g. Thm. 6.1, Prop. 6.2) of multiple stochastic integration (MSI) and related results for the (compensated) Poisson measure , the Gamma measure  and the laws of other compound Poisson type (CPT) processes;

• the intrinsic exponential nature of CPT laws (§§3.1&5.1) results in an expression for their cumulants (eq. (5.2)) in terms of the moments of the associated intensity and Lévy measures and more generally prompts to cumulants as preeminent over moments for computational purposes;

• if the moments of the CPT laws’ Lévy measures have a peculiar factorial part – we discuss the case of  as a prototypical example – the combinatorial complexity of MSI, pertinent to set partition lattices, is reduced to that of permutations (§2.1, Rem. 5.4, Prop. 5.7);

• in the case of , such combinatorial complexity is traced back to the simplicial part of the law, namely the Dirichlet–Ferguson measure , whence to the Dirichlet distribution on the finite-dimensional standard simplex. We show how a detailed study of the latter cannot prescind from combinatorics and algebra and yields interesting connections with enumeration theory (§4.3) and Lie representation theory (§§4.4&8), which we thoroughly explore in the finite-dimensional framework.

Throughout the paper we focus on , and as three major examples. Most of the results concerned with  are proved as generalizations of their analogues for the Dirichlet distribution in finite dimensions. Where such generalizations are not yet at hand, suitable conjectural statements and heuristics are provided (§5.3), on which we plan to base future work along the line sketched in §7.

#### Motivations

Our study is partly motivated by that of chaos representation (§6.1) for square-integrable functionals of CPT random measures, which provide a large class of interesting examples. As noticed in Kondratiev et al. [34, 4.5] for the Gamma measure (§5.2), CPT processes do not posses the chaos representation property (CRP). As a consequence, decompositions of the space of square-integrable functionals obtained via generalized Fourier transforms become in this case particularly involved (cf. e.g. [34, 46, 9]). A useful tool in the ‘simpler’ framework of CRP for processes of Poisson and Gaussian type is Engel–Rota–Wallstrom MSI theory, based on the combinatorial properties of set partitions lattices (see the standard reference monograph Peccati–Taqqu [54]). Despite the great generality achieved and some strikingly simple formulations of otherwise highly non-trivial expressions for moments and cumulants of multiple stochastic integrals (e.g. Rota–Wallstrom [58]), such theory heavily relies on set partitions equations, usually hardly tractable in explicit non-recursive forms.

Partly because of such complexity, the chaos representation for CPT processes has been variously addressed (cf. e.g. [19, 46]) by MSI with respect to power jump processes (i.e. Nualart–Schoutens chaos representation [53]), via orthogonalization techniques for polynomials on distributions spaces [46], through vector valued Gaussian white-noise (Tsilevich–Vershik [72]), or by MSI with respect to Poisson processes, which is mostly the case of our interest. In that framework, unitary isomorphisms are realized, via Jacobi fields, to Fock-type spaces. Whereas the combinatorial properties of such Jacobi fields are often intractable and usually settled by means of recursive descriptions (e.g. [14, 35, 46, 9]), the said extended/non-standard Fock spaces [35, 34, 46, 9, 12] associated to CPT processes display an intrinsic structure, addressed in full generality in Lytvynov [46], and connected to (the enumeration of) set partitions, thus hinting to Engel–Rota–Wallstrom theory. A basic example of this interplay between MSI and set partitions (lattices) in the CPT case is the ‘minimality’ of the embedding [34] of the standard Fock space associated to  into the extended Fock space of  (see Rem. 6.3).

Set partitions enumeration appearing as a tool in the aforementioned approach to CRP turns out to be an incarnation of moments/cumulants dualities for the laws of the considered CPT processes, in which setting we show that cumulants (rather than moments) are the interesting quantities. We base this observation, of rather physical flavor, on the study of Gibbs measures, a Leitmotiv in heuristic derivations in probability from Feynman’s derivation of the Wiener measure to that of the entropic measure in Wasserstein diffusion (von Renesse–Sturm [73]). Indeed, the Helmholtz free energy (see Rem. 5.6) associated to a Gibbs measure is (up to sign and normalization) the cumulant generating function of the energy of the system, and the additivity of the latter translates into a classical feature of cumulants: linearity on independent random variables. This property becomes of particular importance when the partition function of the system, here playing as a moment generating function, may be expressed – in a ‘natural’ way – in exponential form. Physical heuristics aside, in the CPT framework the required exponential form is clear, as it is provided by the well-known Lévy–Khintchine formula (3.3); hence, for example, the cumulant generating function of a CPT random measure with no drift nor Brownian part is linear in the Lévy measure. The computation of cumulants becomes then essentially trivial and moments are recovered via duality.

#### Results

Among other CPT processes, we compare the law  of the Poisson process with that, , of the Gamma process, showing how set partitions combinatorics entailed in the moments/cumulants duality may be significantly simplified, thus making the set partitions lattices machinery unnecessary. Indeed, while the moments of  are given by Bell polynomials (computed at monomial powers, see Thm. 5.2 & Rem. 5.8) – hence, naturally, by averages over set partitions –, those of  are given by augmented cycle index polynomials of symmetric groups (Prop. 5.7) – related in the same way to integer partitions.

On the one hand, provided this understanding of moments, we show how convoluted (recursive) expressions for the scalar product of the extended Fock space of  (and of other non-standard Fock spaces) follow immediately (cf. e.g. Prop. 6.2) from classical identities for Bell polynomials. On the other hand, the correspondence between integer partitions and the cyclic structure of permutations (§2.1) suggests that the arising structure is purely a property of the ‘simplicial part’ (cf. [66]) of , i.e. of the Dirichlet–Ferguson measure  (see [22], §5.2), supported on a space of probability measures.

Alongside with its finite-dimensional analogue, i.e. the Dirichlet distribution (see [52], §4.1), the Dirichlet–Ferguson measure  has been widely studied both per se (e.g. Lijoi–Regazzini [42]) and in connection with the Gamma process (Tsilevich–Vershik–Yor [66, 67]), mostly owing to its numerous applications, ranging from population genetics to coalescent theory, number theory and Bayesian nonparametrics (see e.g. the monograph Feng [21] and the surveys Lijoi–Prünster [41] and Berestycki [11]). In this work, we focus on the combinatorial and algebraic properties of the Dirichlet distribution, which we subsequently extend to the case of .

On the combinatorial side, deepening results by Kerov–Tsilevich [30], we compute the moments of the Dirichlet distribution (Thm. 4.2) as cycle index polynomials of symmetric groups and interpret this result in light of Pólya enumeration theory (Prop. 4.4). As a byproduct, we obtain a new representation (Prop. 4.5) and asymptotic formulae (Prop. 4.6) for the moment generating function of the distribution, i.e. the second multivariate Humbert function , a confluent form of the Lauricella function ; we subsequently generalize this limiting behavior to the Dirichlet–Ferguson measure, matching on arbitrary compact separable spaces a result in [73] for the entropic measure on the 1-sphere (Prop. 5.5). A key rôle in this combinatorial perspective is assumed by the actions of symmetric groups on supports of Dirichlet distributions (i.e. the standard simplices), for which we provide different complementary interpretations (Rem. 4.1, Prop. 8.3).

On the algebraic side, a metaphor of urns and beads (§4.3) points to the dynamical symmetry algebra of the second Humbert function. By means of the general theory in Miller [49, and ref.s therein] we show (Thm. 8.2) that the said algebra is the special linear Lie algebra of square matrices with vanishing trace. The result is then – partly heuristically – extended by means of Lie theory to the infinite-dimensional case of  (Thm. 5.9, Conj. 5.11); the construction is reminiscent of Vershik’s construction of the infinite-dimensional Lebesgue measure [70]. We point out that this connection with Lie theory arising from the study of Dirichlet(–Ferguson) measures is not entirely surprising: For instance, several results concerned with the Gamma measure (e.g. Kondratiev–Lytvynov–Vershik [36, and ref.s therein]) were obtained as part of a much wider program (cf. e.g. [72, §1.4]) to study infinite-dimensional representations of (measurable) -current groups.

We postpone some final remarks to §7.

#### Summary

Firstly, basic facts on partitions and permutations, exponential generating functions and multi-sets are reviewed (§2). Secondly, some notions are recalled on infinitely divisible distributions and the Lévy–Khintchine formula for measures on finite-dimensional linear spaces (§3.1) and on cylindrical measures on nuclear spaces (§3.2), turning then to dualities between moments and cumulants for measures on such spaces (§ 3.3) and their applications to CPT processes. The main results are contained in §§4&5, where we comparatively explore some properties of the Poisson, Gamma and Dirichlet distributions, respectively of the laws  and  of Poisson, Gamma and Dirichlet–Ferguson processes, mostly focusing on the Dirichlet case. The combinatorial properties of (extended) Fock spaces are addressed in §6. Some conclusive remarks are found in §7. Finally, complementary results and proofs, mainly concerned with Lie theory, are collected in the Appendix §8.

## 2 Combinatorial preliminaries

#### Notation

Let be a positive integer (in the following usually implicit) and set

 x\coloneqq (x1,…,xn) ei\coloneqq (10,…,0,i1,0,…,n0) 1\coloneqq (11,…,n1) ixj\coloneqq (xi+1,xi+2,…,xj−1,xj) |x|\coloneqq x1+⋯+xn x^ı\coloneqq (x1,…,xi−1,xi+1,…,xn) →n\coloneqq (1,…,n) x⊕y\coloneqq (x1,…,xn,y1,…,yn) \coloneqq {1,…,n} π∈Sn\coloneqq {permutations of [n]} xπ\coloneqq (xπ(1),…,xπ(n)) x⋄y\coloneqq (x1y1,…,xnyn) x⋄n\coloneqq x⋄⋯⋄xn times x⋅y\coloneqq x1y1+⋯+xnyn,

where we stress the position of an element in a vector with a left subscript. We stress that  is always a signed quantity and the symbol  never denotes a norm of the vector .

For or with , write

 f⋄[x]\coloneqq (f[x1],…,f[xn]) f⋄[x]\coloneqq (f1[x1],…,fn[xn]) f[x]\coloneqq (f⋄[x])1 f[x]\coloneqq (f⋄[x])1,

e.g. while . Finally, denote by the Euler Gamma function, by the Pochhammer symbol of , by

 B[x,y]\coloneqqΓ[x]Γ[y]Γ[x+y] resp. B[x]\coloneqq

the Euler Beta function, resp. its multivariate analogue. We reserve the upright typeset ‘’ for the Beta function, so that no confusion with Bell polynomials or Bell numbers (see below) may arise.

### 2.1 Partitions and permutations combinatorics

We briefly review some basic facts about partitions, permutations and multi-sets. Concerning set partitions, resp. Bell and Touchard polynomials, a more exhaustive exposition may be found in [54, §2.3, resp. §2.4]; however, we rather put the emphasis on the comparison among enumeration of set partitions, integer partitions and permutations. Further related results in enumerative combinatorics may be found in [62].

#### Set and integer partitions

A set partition of is a tuple  of disjoint subsets , termed clusters or blocks, and such that . For any such partition write  and if , i.e. if has clusters.

A (integer) partition  of  into  parts (write: ) is a vector of non-negative integer solutions of the system , . Term a (integer) partition of (write: ) if the second requirement is dropped. We always regard a partition in its frequency representation, i.e. as the vector of its ordered frequencies (see e.g. [2, §1.1]).

To a set partition one can associate in a unique way the partition by setting . The number of set partitions with subsets of given cardinalities is counted by the Faà di Bruno’s coefficient (or multinomial number of the third kind)

 M3[λ]\coloneqqn!(n∏i(i!)λiλi!)−1=n!λ!(→n!⋄)λ.

An interpretation of  in terms of set partitions lattices may be found in [54, (2.3.8)].

A permutations is said to have cyclic structure  if the lengths of its cycles coincide with (a permutation is thus always understood in its one-line notation). Denote by  the set of permutations with cyclic structure  and recall that two permutations have the same cyclic structure if and only if they are conjugate to each other as group elements of . Also recall (cf. [62, I.1.3.2]) that

 M2[λ]\coloneqq#Sn(λ)=n!(n∏iiλiλi!)−1=n!λ!→nλ,

termed multinomial number of the second kind. Given , denote by the unique integer partition of encoding the cycle structure of , so that .

Finally, a multi-index of size and length is any vector in such that . Such multi-indices encode the functions by regarding as . Their count is given by the multinomial coefficient  of the multi-index.

In a similar fashion, a positive multi-index of size and length is any vector in such that . Such multi-indices encode surjective functions in , since (strict) positivity of every entry in entails that every element of is targeted by . Up to permutation of the elements of , surjective functions in are bijective to set partitions of into blocks, hence the latter ones’ total count is given by .

###### Remark 2.1.

For the sake of completeness, let us point out that – in this context – the multinomial coefficient ought to be thought of as the first element in a sequence  and deserves the name of multinomial number of the first kind. We prefer however to keep the usual notation and terminology, but we shall say ‘multinomial numbers’ when collectively referring to the three of them.

As well as being encoded by a partition , the cyclic structure of a permutation  with  cycles is also encoded by a multi-index of length . Visually, the set of such multi-indices is bijective to that of monotone excursions on the lattice starting at and ending at . In this case, counts the number of non-zero elements in , which in turn count the lengths of cycles. The general statement is readily deduced from the following example via a Catalan-type diagram

The horizontal lines describe the cycles of the following permutation, resp. partition, multi-index

 π= (⋅)(⋅⋅)(⋅⋅⋅⋅)(⋅)(⋅⋅⋅)∈S11, λ= (12,21,31,41,50,…,110)⊢511 m= (1,2,0,4,0,1,0,0,3)∈N90.

The latter is obtained by counting the length of horizontal lines on the lower side of each square, separated by as many ’s as the number of vertical lines on the right side of each square, reduced by one. The number of parts in which is subdivided, given by , accounts for the number of non-zero entries in , hence for the number of different groups of contiguous horizontal lines in the diagram.

###### Remark 2.2.

When regarded as encoding the cyclic structure of a permutation, multi-indices and integer partitions ought to be understood as ‘dual’ notions, the ‘duality’ being given by flipping (along the main diagonal) the Catalan type diagram for  constructed above. Loosely speaking, if a property involving a multi-index  in  and of length  holds, then one would expect some ‘dual’ property to hold for the partition corresponding to . As the correspondence between integer partitions and multi-indices is not bijective – hence the quotation marks –, we shall refrain from addressing it further. It will however be of guidance in discussing instances of the aggregation property of the Dirichlet distribution in §4 below.

#### Bell polynomials and the cycle index polynomial of Sn

The cluster structure of a partition , resp. , is encoded by the partial, resp. complete, Bell polynomial

 Bnr[x]\coloneqq ∑λ⊢rnM3[λ]xλ, resp. Bn[x]\coloneqq ∑λ⊢nM3[λ]xλ, (2.1)

where and the index of each variable in the monomial indicates the size of the cluster, i.e. there are  clusters of size , up to clusters of size . Bell polynomials count the Bell number of partitions of  and satisfy the identities

 Bnr[b(a1)⋄→n⋄x]= anbrBnr[x] Bn[(a1)⋄→n⋄x]= anBn[x], (2.2)

the recursive identity

 Bn+1[x]= n∑k=0(nk)Bn−k[xn−k]xk+1 B0[∅]\coloneqq 1, xℓ\coloneqq (x1,…,xℓ), (2.3)

and the binomial type (cf. [57]) identity

 Bn[x+y]= n∑k=0(nk)Bk[xk]Bn−k[yn−k]. (2.4)

In the same way as the Bell polynomial encodes (the cardinalities of) set partitions of , the cycle index polynomial of  defined by

 Zn[x]\coloneqq Bn[Γ⋄[→n]⋄x]n!=1n!∑λ⊢nM2[λ]xλ (2.5)

encodes integer partitions of and satisfies the recurrence relation

 (n+1)!Zn[x]=n∑k=0n!Zn−k[xn−k]xk+1 Z0[∅]\coloneqq1, xℓ\coloneqq (x1,…,xℓ). (2.6)
###### Remark 2.3 (Cycle index polynomials of permutation groups).

Cycle index polynomials may be defined for any permutation group  by the formula

 ZG[x]\coloneqq1#G∑π∈Gxλ(π).

One of their numerous applications is Pólya Enumeration Theory, whose main result is recalled in §4.3 below.

A combinatorial proof of (2.3) and (2.6) is best given in terms of Young tableaux. Indeed, let e.g. , which induces by augmentation the following partitions of

 λ=\Yvcentermath1\yng(4,4,2,1)⟶\Yvcentermath1\yng(5,4,2,1)\yng(4,4,3,1)\yng(4,4,2,2)\yng(4,4,2,1,1),

obtained by adding to each subset of one additional element in such a way that the associated tableau remains of Young type. Labelling each box in a tableau allows to consider set partitions rather than integer partitions.

For arbitrary , the set partitions of are thus obtained by fixing  a subset of  elements and adding to it the additional element , so that the size of the resulting cluster is encoded in the variable , while the partitions of the remaining  elements are encoded in the Bell polynomial . Since there are  possible choices for the subset , equation (2.3) follows. Taking this choice to be irrelevant, for each of the so chosen clusters has cardinality , we get (2.6).

### 2.2 Exponential generating functions

Given a sequence of real numbers, we denote by its exponential generating function.

#### EGF’s and partitions

It is a well-known fact in enumerative combinatorics that exponential generating functions provide yet another way to account for multinomial numbers, hence to count e.g. set partitions.

###### Proposition 2.4.

Let and for . It holds

 k∏ℓfℓ[x]= EGF[an](x) an\coloneqq ∑m∈Nk0|m|=n(nm)a(1)m1⋯a(k)mk (2.7a) f[x]kk!= EGF[bn](x) bn\coloneqq ∑m∈Nk1|m|=n1k!(nm)am1⋯amk (2.7b) ef[x]= EGF[cn](x) cn\coloneqq ∑λ⊢nM3[λ]aλ11⋯aλnn=Bn[a1,…,an]. (2.7c)
###### Proof.

A proof of (2.7a) (whence of (2.7b)) follows by induction from the classical Cauchy product of power series. Proofs of (2.7a) and of the formula for the composition of EGF’s (whence of (2.7c)) are also found in [62, 5.1.3, 5.1.4], together with the respective combinatorial interpretations. We rather discuss the specific combinatorial interpretation of (2.7c), which is as follows. Firstly, we have from (2.7b)

 ef[x]=∞∑k=0f[x]kk!=∞∑k=0∑m∈Nk1|m|=n1k!(nm)am1⋯amk.

As shown in the last paragraph, the coefficient  in (2.7b) accounts for the number of surjective functions in , which we can regard as set partitions of . Since  is itself varying in , we are interested in set partitions  into clusters of arbitrary cardinality (hence in set partitions with arbitrary number of clusters). Letting the subscript  of  denote the cardinality of the set  in , the number of clusters in  with cardinality  is then given by coefficient  in the integer partition , whence (2.7c) follows by the above discussion on Bell polynomials. ∎

###### Example 2.5.

The exponential generating function for the number of cyclic permutations of order , given by , is

 EGF[M2[en]](x)=∑nΓ[n]xnn!=−log(1−x). (2.8)

As a simple check, since every permutation is uniquely partitioned into cyclic permutations of maximal length, the exponential generating function for the total number of permutations is obtained by exponentiating the one above, viz.

 EGF[#Sn](x)=11−x=∞∑n=0n!xnn!.

### 2.3 Multi-sets

We conclude this section by recalling some basic properties of multi-sets.

#### Notation

Given a set , a finite -multi-set is any function such that is finite, where  denotes integration on with respect to the counting measure. We are mainly interested in real multi-sets, which we denoted by

 {s1,…,s1α1,s2,…,s2α2,…}\eqqcolon\llbracketsα\rrbracket,

where  is the vector of positive values attained by the function  and  is a real valued vector with mutually different entries; since is finite,  is a simple function and thus has finitely many entries; furthermore .

We term the set  the underlying set to , the number  the multiplicity of  the number of (different) types in  and  the cardinality of the multi-set. Finally, recall that the total number of multi-sets of cardinality  with  types is given by the number of solutions in  of the equation and counted (see [62, §I.1.2]) by the multi-set coefficient

For further purposes notice that the expression  is meaningful for any real .

## 3 Moments and cumulants on nuclear spaces

We give here a short review of moments and cumulants in fairly broad generality, namely for (probability) measures on nuclear dual spaces (i.e. nuclear spaces admitting a strongly continuous pre-dual, not to be confused with dual nuclear or co-nuclear spaces, i.e. spaces with nuclear dual) and more generally for cylindrical measures on topological linear spaces.

### 3.1 Infinitely divisible distributions and the Lévy–Khintchine formula

For the sake of simplicity, we review here Lévy processes on , mainly following [59, 3, 63]. Generalizations to measure-valued Lévy processes on Riemannian manifolds, Lévy processes on (separable) Banach and nuclear spaces may be found in [46][4] and in [68] respectively. For convolution and infinite divisibility of measures on finite-dimensional spaces see [59, §7] or [63, §3.2.1]; for the convolution calculus on spaces of distributions with applications to stochastic differential equations see e.g. [10]. For those Lévy processes on Polish spaces relevant in the applications to random measures below specific references are provided in §5.

#### Infinite divisibility

A probability distribution on is termed infinitely divisible if for every positive integer  there exists a probability distribution  such that , where  denotes the -fold convolution of the measure  with itself and we set . In words, a distribution  is infinitely divisible if it has convolution roots of arbitrary integer order in the convolution algebra of measures on , in which case each convolution root is unique (see [59, 7.5-6]) and it is possible to define arbitrary (non-integer) convolution powers (see [59, §7]) of . Term further a (probability) distribution  on  to be of Poisson type if there exists  and a probability distribution , defined on the same space as , such that

 ν=νcλ=exp∗[cλ]\coloneqqe−c∞∑n=0ckk!λ∗k, (3.1)

where  is a normalization constant. We term  the convolution logarithm of  and denote it by . Morally, a distribution is infinitely divisible if and only if it admits a well-defined convolution logarithm; it is indeed possible to show (see [63, §3.2.1]) that every infinitely divisible distribution is in the closure of Poisson type measures with respect to the narrow topology.

#### Lévy processes

Recall that a Lévy process in law on  is any stochastically continuous process starting at  with stationary independent increments; any such process admits a modification with càdlàg paths (see [3, 2.1.7]), usually termed a Lévy process; since we are mainly interested in the laws of Lévy processes, the term ‘in law’ is henceforth omitted and, whenever relevant, only modifications with càdlàg paths are considered. A distribution on is the law of some Lévy process if and only if it is infinitely divisible (see [59, 7.10]) and distributions as such are characterized by the following well-known representation theorem.

###### Theorem 3.1 (Lévy–Khintchine formula (see e.g. [59, 8.1])).

A probability distribution  on  is infinitely divisible if and only if there exist

• a constant  in ;

• a non-negative definite quadratic form  on ;

• a Lévy measure  on , i.e. satisfying

 λ{0d}= 0 and ∫Rd(1∧∥x∥22)dλ(x)< ∞, (3.2)

such that

 F[ν](z)=exp[−12A[z,z]+ic⋅z+∫Rd(eiz⋅x−1−i(z⋅x)\mathds1D(x))dλ(x)], (3.3)

where and denotes the closed ball in of radius , centered at the origin. Furthermore, if this is the case, then , and are unique.

It is well-known (see e.g. [59, 2.5&7.5]) that the Fourier transform of an infinitely divisible distribution on  is a non-negative definite (in the sense of (3.4) below) functional on , continuous at  with value , and nowhere vanishing. Whereas these conditions are not sufficient to ensure infinite divisibility, we are interested in those infinite-dimensional linear spaces (playing the rôle of ) such that any functional as above is the Fourier transform of some probability measure on . A wide class of such spaces, seemingly sufficiently general to include most applications, is constituted by nuclear spaces.

### 3.2 Measures on nuclear spaces

We follow [25, §IV.1.1-4, IV.4.1-2]. A modern treatment of analysis on nuclear spaces may be found in [13]. Topological linear spaces are always meant to be Hausdorff; the term is however omitted.

#### Cylindrical measures

Let be a co-nuclear space and denote by its topological dual (a nuclear dual space) endowed with the weak* topology induced by the canonical duality pair . Let further  be a finite-dimensional linear subspace of  and denote by  its annihilator in . Given a subset  of the finite-dimensional vector space  define the cylinder set with base set and generating subspace  as the pre-image of under the quotient map . Since is finite-dimensional, it may be endowed with a unique (locally convex) vector space topology and the induced Borel -algebra. By a cylindrical measure  on  we mean a non-negative real-valued function on the cylinder sets of , countably additive on families of cylinder sets with same generating subspace and disjoint Borel measurable base sets, additionally satisfying the normalization . A cylindrical measure is termed continuous if the functionals

 ⟨u∘y⟩ν\coloneqq∫Φ′u[y⋄(h1)]dν(h)

are sequentially continuous in the variables on for every real-valued function  continuous and bounded on  and arbitrary in . With common abuse of notation, we hereby implicitly denote by the linear functional for  in , hence we set

 y⋄(h1)\coloneqq(y1h,…,ykh)=(⟨h|y1⟩,…,⟨h|yk⟩),

and by  the expectation with respect to . Any continuous cylindrical measure on a nuclear dual space is countably additive (see [25, §IV.2.3]), thus in fact a probability measure.

#### Fourier & Laplace transforms

Given a cyclindrical measure on , define its Fourier, resp. Laplace, transform in the variable  as

 F[ν](y)\coloneqq ⟨expiy⟩ν, resp. L[ν](y)\coloneqq ⟨expy⟩ν.

We borrow the term transform from infinite-dimensional analysis; in the case  those of characteristic, resp. moment generating, function are of common use. It is not difficult to show (see [25, §IV.4.1]) that the Fourier (or Laplace) transform of is actually characterized by the -measure of half-spaces in . Indeed, for any finite-dimensional subspace , every half-space of the form with in and a real constant consists of cosets induced by the quotient map (see above). Thus, the -measure of the half-space  in  coincides with the -measure of the half-space defined by the same inequality in the quotient space  and it holds that

 F[ν](y)=F[pr♯ν](y).

#### The Bochner–Minlos Theorem

Among the reasons why we are interested in cylindrical measures on nuclear spaces is the following well-known realization theorem, which allows for the identification of those linear functionals on a space  that are Fourier transforms of some (cylindrical) measure on the dual space . Recall that a functional  on a topological linear space  is termed non-negative definite if

 k∑i,jF(yi−yj)αi¯¯¯¯αj≥0 (3.4)

for every in  in  and  in .

###### Theorem 3.2 (Bochner–Minlos (see [25, §iv.4.2])).

Let be a functional on a topological linear space . Then, is the Fourier transform of a cylindrical measure  on  if and only if it is non-negative definite, sequentially continuous and such that .

If, in addition, is a nuclear space (hence is a nuclear dual space), the same statement holds for (countably additive, rather than only cylindrical) probability measures.

### 3.3 Moments, central moments and cumulants

We are solely concerned with linear moments and cumulants, i.e. those of linear functionals. However, when referring to moments and cumulants, the term ‘linear’ is always omitted.

#### Notation

Let be a cylindrical measure on a (topological) dual linear space . By (raw), resp. central, (univariate) moments of we mean any integral of the form

 μ′νn[y]\coloneqq ⟨yn⟩ν, resp. μνn[y]\coloneqq where yn\coloneqq⟨⋅|y⟩n for fixed y in Φ. We drop the superscript ν whenever the measure may be inferred from the context, thus writing only μn in place of μνn. Let now y\coloneqq(y1,…,yk) be a vector in Φ×k and denote the vector in the same way. The multivariate Fourier, resp. Laplace, transform in the variables y is defined as F[ν](y)\coloneqq resp. L[ν](y)\coloneqq The corresponding multivariate moments of order n are defined analogously by setting μ′νm[y]\coloneqq ⟨ym⟩ν, resp. μνm[y]\coloneqq ⟨(y−⟨y⟩⋄ν)m⟩ν, where m is any multi-index in Nk0 of length n and, consistently with the established notation and the aforementioned abuse thereof, we set ⟨w⟩ν\coloneqq ⟨w1⟩ν⋯⟨yk⟩ν, and ⟨w⟩⋄ν\coloneqq (⟨w1⟩ν,…,⟨wk⟩ν).

Whenever (whence ), we write in place of in order to keep track of the dimension of . Finally, we denote collectively by both raw and central moments. Given as above and , denote by the product . We define further the total multivariate cumulant of order as (cf. [54, 3.2.19])

 (3.5)

the corresponding univariate version being simply

 κn[y]\coloneqqκn[y1].

More general definitions of cumulants are possible, which can be straightforwardly adapted to the present setting. For instance (see e.g. [44, §A]), one can recursively define multivariate cumulants with arbitrary indices in terms of lower order moments by setting

 κL[y]\coloneqq κ∅[y]\coloneqq 0, (3.6)

in which case .

Finally, cumulants enjoy the following properties

1. homogeneity: for  in ;

2. additivity: if , then ;

3. shift equivariance: for ;

4. shift invariance: for  and ;

5. independence: as soon as for some non-trivial .

The same holds in fact for an arbitrary measure  and random variables  (see e.g. [54, §3.1] for the easy proofs).

#### Dualities

We stress that, a priori, none of the above integrals is well-defined.

If the (-variate) Laplace transform is well-defined and strictly positive on some domain (resp. ), one can define the cumulant generating function of by setting, on the same domain

 K[ν]\coloneqqlnL[ν].

In light of the discussion on half-spaces in §3.2, we shall focus on (cylindrical) measures on such that  and  (resp.  and ) are analytic in the variable(s)  (resp. ) in a neighborhood of  (resp. in ) for every fixed in some affine open half-space  containing  (resp. in some affine hyper-octant of ). We shall term these measures to be analytically of exponential type, in which case one can check by straightforward computations that the (multivariate) raw moments, resp. (multivariate) cumulants, are precisely the coefficients in the McLaurin expansion of the respective generating functions; that is e.g.

 μ′m[y]= (−i)|m|∂mt∣∣t=0F[ν](t⋅y) t∈Rk, (3.7a) κm[y]= (−i)|m|∂mt∣∣t=0lnF[ν](t⋅y) t∈Rk, (3.7b)

where we denote by the multivariate differential operator of indices in the variables . The identity (3.7b) (usually taken as a definition) yields the cumulance property, motivating the terminology. Namely, whenever  are finite Borel measures on a topological group with analytic Fourier transform and such that their convolution is defined, then by properties of the latter

 K[ν1∗ν2]=K[ν1]+K[ν2]

so that .

###### Theorem 3.3 (Moments/cumulants and central/raw moments dualities).

Let  be a (cylindrical) measure analytically of exponential type on a (topological) dual linear space . Then, the moments and the cumulants  may be inferred from each other via the formulae

 μ′n= Bn[κ1,…,κn], (3.8a) κn= B∗n[μ′1,…,μ′n]\coloneqqn∑k=1(−1)n−k(n−k)!Bnk[μ′1,…,μ′k], (3.8b)

where and are the Bell polynomials (2.1). Analogously, the raw moments and the central moments are related by the binomial transforms (also Möbius inversion formulae [54])

 μn= n∑k=0(nk)(−1)n−kμ′n−k1μ′k μ′0= 1, (3.9a) μ′n= n∑k=0(nk)μ′n−k1μk μ0= 1,μ1=0. (3.9b) The corresponding (recursive) multivariate formulae for moments and cumulants are straightforwardly deduced from (3.5)-(3.6), while those for raw and central moments are as follows μm[y]= ∑0≤j≤m(mj)(−1)|m−j|⟨y⟩m−jνμ′j[y] μ′0= 1, (3.9c) μ′m[y]= ∑0≤j≤m(mj)⟨y⟩m−jνμ