COMBINATORIAL LEVY PROCESSES

HARRY CRANE

Abstract. Combinatorial Levy processes evolve on general state spaces of countable combi-natorial structures. In this setting, the usual Levy process properties of stationary, indepen-dent increments are defined in an unconventional way in terms of the symmetric differenceoperation on sets. In discrete time, the description of combinatorial Levy processes gives riseto the notion of combinatorial random walk. These processes behave differently than randomwalks and Levy processes on other state spaces. Standard examples include processes onsubsets of a countable set, graphs with countably many vertices, and n-ary relations, but theframework permits far more general possibilities. The main theorems characterize combi-natorial Levy processes by a unique σ-finite measure. Under the additional assumption ofexchangeability, we obtain an explicit Levy–Ito–Khintchine-type characterization, by whichevery exchangeable combinatorial Levy process corresponds to a Poisson point process onthe same state space.

1. Introduction

A Levy process on Rd is a random path t 7→ Xt with stationary, independent incrementsand cadlag sample paths with respect to the Euclidean topology. Levy processes comprisea large class of tractable models with applications in finance, neuroscience, climate mod-eling, etc., and the Levy–Ito–Khintchine theorem decomposes their rich structure into anindependent Brownian motion with drift, a compound Poisson process, and a pure jumpmartingale. Properties of R-valued Levy processes specialize those of Levy processes ingeneral topological groups; see Bertoin [8] for a survey of the real-valued setting. In anarbitrary topological groupX, the Levy process assumptions are defined with respect to thegroup action, with the left, respectively right, increment between x, x′ ∈ X defined as theunique y ∈ X such that x = yx′, respectively x = x′y. Liao [28] gives a general introductionto Levy processes in topological groups with special treatment of the Lie group case, whichgarners special interest for its relation to certain types of stochastic flows.

In both the real-valued and Lie group setting, many nice properties result from theinterplay between the increments assumptions and the topology of the underlying statespace. In Euclidean space, the Levy–Ito–Khintchine representation is tied to its predecessor,the Levy–Khintchine theorem for infinitely divisible distributions. In a Lie group, thesmoothness of the associated Lie algebra plays a key role.

Afield of Levy processes, combinatorial stochastic processes evolve on discrete statespaces, with a focus on the theory of exchangeable random partitions [15, 19, 26], coalescentand fragmentation processes [7, 9, 27, 31], connections to stable subordinators, Brownianbridges, and Levy processes [30, 33], tree- [1, 2, 3, 12, 18, 32] and graph-valued [14, 16]

Date: March 8, 2016.1991 Mathematics Subject Classification. 60J25; 60G09; 60B15.Key words and phrases. combinatorial stochastic process; Levy process; dynamic networks; Levy–Ito–

Khintchine representation; exchangeability.1

2 HARRY CRANE

processes. In applications, these processes serve as models for dynamic structures that arisein streaming data collection and complex network applications. Adding a temporal com-ponent to the already complicated structural features of combinatorial objects introducesa potentially intractable amount of flexibility that invites arbitrary dynamics and abstrusedescriptions. We introduce combinatorial Levy processes as a reasonable family of models forthis purpose.

Combinatorial Levy processes evolve on discrete spaces of labeled combinatorial objects.Rather than restrict attention to any specific state space, we develop a theory that encom-passes special cases as well as more general processes. The following special cases motivateour general treatment.

• Set-valued processes: On the space of subsets of N, a combinatorial Levy processevolves by rearranging elements. For example, each element i = 1, 2, . . . mightenter and leave the set at alternating times of independent rate-1 Poisson pointprocesses. These dynamics imitate those of some previously studied partition-valued processes [10, 13, 30]. Forty years ago, Harris [21, 22] studied set-valuedprocesses under entirely different assumptions.• Graph-valued processes: Perhaps the most immediate contemporary interest in com-

binatorial Levy processes is in modeling dynamic networks. Even given the recentinterest in complex networks, stochastic process models for dynamic networks havereceived little attention: we know of only [23, 25, 34] in the statistics literature,[14, 16] in the probability literature, and [20] and some follow up articles in theepidemiology and physics literature. Our main discussion explicitly describes thepossibilities and limitations of combinatorial Levy process models for dynamicnetworks. Our main theorem stratifies the behavior of graph-valued Levy processesinto a hierarchy of global, vertex, and edge-level discontinuities in a parallel mannerto the Aldous–Hoover decomposition of exchangeable random graphs [4, 24].• Networks with community structure: Another interesting place for this theory is in

modeling composite structures, such as dynamic networks with an underlyingcommunity structure. In this case, it is natural to combine the above two processeson sets and graphs into a process that models the joint evolution of a network and acommunity of its vertices. Extensions to collections of k different communities and ldifferent networks and possibly higher order interactions also fall within the scopeof combinatorial Levy processes.

1.1. Outline. In Section 2, we summarize the main theorems in the case of set-valued Levyprocesses. In Section 3, we lay down key definitions, notation, and observations. In Section4, we formally summarize the main theorems in the language of Section 3. In Section 5,we demonstrate our main theorems with concrete examples that are relevant to specificapplications. In Section 6, we prove a key theorem about σ-finite measures on combina-torial spaces, from which we readily deduce the Levy–Ito–Khintchine representation forexchangeable combinatorial Levy processes. In Section 7, we prove our main theorems. InSection 8, we make concluding remarks.

2. Exposition: set-valued processes

Remark 2.1 (Notation). We discuss both discrete and continuous time processes. When speakinggenerally, we index time by t ∈ [0,∞). When speaking specifically about discrete time processes, weindex time by m ∈ Z+ := {0, 1, . . .} and write X = (Xm, m ≥ 0).

COMBINATORIAL LEVY PROCESSES 3

We introduce the concept of combinatorial increments to capture structural differencesbetween combinatorial objects. To fix ideas, we first assume X = (Xt, t ≥ 0) evolves on thespace of subsets of a base set S ⊆N, denoted 2S.

2.1. Increments and topology. Every A ⊆ N determines a map A : 2N → 2N by A′ 7→A4A′, where

(1) A4A′ := (A ∩ A′c) ∪ (Ac∩ A′)

is the symmetric difference operation and Ac :=N \A denotes the complement of A. Under thisoperation, the empty set ∅ := {} acts as the identity and each A ⊆N is its own inverse, thatis, A4A = ∅ for all A ⊆N. We equip 2N with the product discrete topology induced by

(2) d(A,A′) := 1/(1 + sup{n ∈N : A ∩ [n] = A′ ∩ [n]}), A,A′ ⊆N,

with the convention 1/∞ = 0.In the following definition, T stands for either discrete time (T = Z+) or continuous time

(T = [0,∞)). The definition holds in either case, the only difference being that cadlag pathsare automatic in discrete time.

Definition 2.2 (Combinatorial Levy process on 2N). We call X = (Xt, t ∈ T) a combinatorialLevy process on 2N if it has

• X0 = ∅,• stationary increments, that is, Xt+s 4Xs =DXt for all s, t ≥ 0, where =D denotes equality

in law,• independent increments, that is, Xt1 4Xt0 , . . . ,Xtk 4Xtk−1 are independent for all 0 ≤

t0 ≤ t1 ≤ · · · tk < ∞ in T, and• cadlag sample paths, that is, t 7→ Xt is right continuous and has left limits under the

topology induced by (2).

We can interpret discrete time combinatorial Levy processes on 2N as set-valued randomwalks.

Definition 2.3 (Set-valued random walk). A random walk on 2N with increment distributionµ onLN and initial state X0 is a discrete time process X = (Xm, m ≥ 0) with Xm+1 =DXm 4 ∆m+1for every m ≥ 0, where ∆1,∆2, . . . are independent and identically distributed (i.i.d.) according to µ.

Theorem 2.4. Let X = (Xm, m ≥ 0) be a discrete time combinatorial Levy process on 2N . Thenthere exists a unique probability measure µ on 2N such that X is distributed as a random walk withinitial state ∅ and increment distribution µ.

The proof of Theorem 2.4 is straightforward even for general combinatorial Levyprocesses—see Theorem 4.4—but we explicitly prove the set-valued case to aid our moregeneral discussion later on.

Proof. The stationary and independent increments assumptions imply that X = (Xm, m ≥ 0)is determined by its initial state X0 = ∅ and an independent, identically distributed sequence∆ = (∆m, m ≥ 1) of subsets, where

∆m = Xm 4Xm−1, m ≥ 1.

For each m ≥ 1, ∆m contains all elements whose status changes between times m − 1 and m;thus, the transition law of X is governed by a unique probability measure µ on 2N , whichacts as the increments measure for the random walk started at ∅. �

4 HARRY CRANE

In continuous time, X = (Xt, t ≥ 0) can experience infinitely many jumps in boundedtime intervals, but cadlag sample paths constrain each induced finite state space processX[n] := (Xt ∩ [n], t ≥ 0) to jump only finitely often in bounded intervals. These competingnotions harness the behavior of X and lead to the following general characterization.

Theorem 2.5. Let X = (Xt, t ≥ 0) be a continuous time combinatorial Levy process on 2N . Thenthere is a unique measure µ on 2N satisfying

(3) µ({∅}) = 0 and µ({A ∈ 2N : A ∩ [n] , ∅}) < ∞ for all n ∈N

such that the infinitesimal jump rates of X satisfy

limt↓0

1tP{Xt ∈ d∆} = µ(d∆), ∆ ∈ 2N \ {∅}.

We observe that any combinatorial Levy process has the Feller property (Corollary 4.7)and, thus, its evolution is determined by the infinitesimal jump rate

µ(d∆) = limt↓0

1tP{Xt ∈ d∆}, ∆ ∈ 2N \ {∅}.

Since ∆ = ∅ corresponds to no jump, we may tacitly assume µ({∅}) = 0. To ensure that eachX[n] jumps only finitely often, µ must also satisfy

µ({A ∈ 2N : A ∩ [n] , ∅}) < ∞ for all n ∈N .

Since the behavior of X is determined by the infinitesimal jump rates limt↓0 t−1P{Xt ∈ d∆},X can be described by a unique measure µ on 2N that satisfies (3).

From any µ satisfying (3), we construct the µ-canonical Levy process X∗µ = (X∗t , t ≥ 0) froma Poisson point process ∆∗ = {(t,∆t)} ⊆ [0,∞) × 2N with intensity measure dt ⊗ µ, where dtdenotes Lebesgue measure on [0,∞). The atoms of ∆∗ determine the jumps of X∗µ and thelaw of X∗µ coincides with the law of X through the following explicit construction. Given

∆∗ and n ∈N, we construct X∗[n]µ = (X∗[n]

t , t ≥ 0) on 2[n] by

• X∗[n]0 = ∅,

• X∗[n]t = X∗[n]

t− 4(∆t ∩ [n]), if (t,∆t) is an atom of ∆∗, and• X∗[n]

t = X∗[n]t− := lims↑t X∗[n]

s if t > 0, is not an atom time in ∆∗, that is, X∗[n] is constantbetween the atom times of ∆∗.

Every combinatorial Levy process admits a canonical version and the spirit of theLevy–Ito–Khintchine theorem lives on; but, rather than the three part decomposition ofLevy–Ito–Khintchine, we observe a correspondence between combinatorial Levy processeson 2N and Poisson point processes with intensity dt ⊗ µ for µ satisfying (3). Theorem 4.5covers the corresponding description of general combinatorial Levy processes.

2.2. Exchangeable processes. For processes X = (Xt, t ≥ 0) and X′ = (X′t , t ≥ 0), we writeX =D X′ to denote that X and X′ have the same finite-dimensional distributions, that is,

(Xt1 , . . . ,Xtr) =D(X′t1, . . . ,X′tr

) for all 0 < t1 < · · · < tr < ∞.

For A ⊆N and any permutation σ :N→N, we denote the relabeling of A by σ by Aσ, where

i ∈ Aσ if and only if σ(i) ∈ A.

COMBINATORIAL LEVY PROCESSES 5

We call X exchangeable if X =D Xσ = (Xσt , t ≥ 0) for all permutations σ : N → N that fix all

but finitely many elements ofN.By Theorem 2.4, the discrete time increments of X = (Xm, m ≥ 0) are independent

and identically distributed from a probability measure µ on 2N . Under the additionalassumption that X is exchangeable, µ must also be exchangeable in the sense that

µ({A∗ ⊆N : A∗ ∩ [n] = A}) = µ({A∗ ⊆N : A∗ ∩ [n] = Aσ}), A ⊆ [n],

for all permutations σ : [n]→ [n], for all n ∈N. Any probability measure ν on [0, 1] inducesan exchangeable measure ν∗ on 2N by

(4) ν∗({A∗ ∈ 2N : A∗ ∩ [n] = A}) =

∫[0,1]

p|A|(1 − p)n−|A|ν(dp), A ⊆ [n], n ∈N,

where |A| denotes the cardinality of A ⊆ [n]. de Finetti’s theorem [17] gives the converse:every exchangeable probability measure µ corresponds to a unique probability measure νon [0, 1] so that µ = ν∗, that is, µ is the ν-mixture defined in (4).

In continuous time, µ in (3) decomposes into mutually singular pieces, invoking aLevy–Ito-type interpretation.

Theorem 2.6. Let X = (Xt, t ≥ 0) be an exchangeable combinatorial Levy process on 2N . Thenthere exists a unique measure ν on [0, 1] satisfying

(5) ν({0}) = 0 and∫

[0,1]s ν(ds) < ∞

and a unique constant c ≥ 0 such that X =D X∗µ, the µ-canonical Levy process defined above with

(6) µ = ν∗ + c∞∑

i=1

εi,

where ν∗ is defined as in (4), with ν now possibly an infinite measure, and εi is the unit mass at{i} ⊂N for each i ∈N.

We call (6) the Levy–Ito–Khintchine representation. See Theorem 4.14 for the generalstatement.

2.3. Projecting into [0, 1]. We can project X = (Xm, m ≥ 0) into [0, 1] by Xm 7→ π(Xm), where

(7) π(Xm) := limn→∞

n−1|Xm ∩ [n]|

is the limiting frequency of elements in Xm. By de Finetti’s theorem and the law oflarge numbers, π(X) := (π(Xm), m ≥ 0) exists almost surely whenever X is exchangeable.Furthermore, by independence of Xm−1 and ∆m, we observe

π(Xm) =D π(Xm−1)(1 − π(∆m)) + (1 − π(Xm−1))π(∆m),

so that π(X) is also a Markov chain on [0, 1]. In continuous time, the projected process((π(Xt), 1−π(Xt)), t ≥ 0) exists almost surely and exhibits the Feller property in the Euclideantopology on the 1-dimensional simplex.

6 HARRY CRANE

2.4. Extending the set-valued case. When moving beyond the set-valued case, the projec-tion operation π : 2N → [0, 1] must be replaced by the more technically involved notion ofa combinatorial limit ‖ · ‖, which maps an object A to an exchangeable probability measure‖A‖ on the space inhabited by A.

In the case where A ⊆ N, we define ‖A‖ as follows. For any injection ϕ : [m]→ N andA ⊆N, we define Aϕ

⊆ [m] by

i ∈ Aϕ if and only if ϕ(i) ∈ A.

For any finite subset S ⊆ [m], we define the limiting density of S in A by

δ(S; A) := limn→∞

1n↓m

∑injections ϕ:[m]→[n]

1{Aϕ = S}, if it exists,

where n↓m := n(n−1) · · · (n−m+1) and 1{·} is the indicator function of the event described by·. (As we discuss later, existence of δ(S; A) is guaranteed whenever A is the realization of anexchangeable random set, and so we do not worry about existence for now.) Together thecollection (δ(S; A), S ∈

⋃m∈N 2[m]) determines a unique, exchangeable probability measure

µ on 2N withµ({A∗ ∈ 2N : A∗ ∩ [m] = S}) = δ(S; A), S ⊆ [m].

We denote this probability measure by ‖A‖.In the set-valued case, ‖A‖ and π(A) encode the same probability measure by noting that

π(A) = p implies

‖A‖({A∗ ∈ 2N : A∗ ∩ [m] = S}) = p|S|(1 − p)m−|S|, S ⊆ [m].

This equivalence is not obvious, but it follows directly from de Finetti’s theorem. There isno such simplification for general structures, and so we must resort to the more technicaldefinition of ‖A‖ in terms of the limiting densities δ(S; A), which we introduce formally inSection 3.2.

Our main theorems lift the foregoing ideas for set-valued processes to Levy processes oncountable combinatorial objects, which no longer have the simple 1-dimensional structure ofsubsets and, thus, require more care. The upshot of our general treatment is an overarchingtheory for modeling dynamic combinatorial structures. A potential liability of this generalframework is that some readers may lose track of the intuition that the main theoremsprovide in special cases. To avoid these pitfalls, we frame our main theorems in the contextof the more tangible cases of set- and graph-valued processes, and we continually revisitthese examples throughout.

3. Combinatorial structures

Remark 3.1 (Notation). We employ the usual notation (x1, . . . , xn) and {x1, . . . , xn} to denoteordered and unordered sets, respectively.

The above examples are special cases of what we call combinatorial structures.

Definition 3.2 (Combinatorial structures). A signature L is a finite list (i1, . . . , ik) of non-negative integers for which 0 ≤ i1 ≤ · · · ≤ ik and ik ≥ 1. Given a signature L = (i1, . . . , ik) and aset S, a combinatorial structure with signature L over S is a collection M = (S; M1, . . . ,Mk)such that M j ⊆ Si j for every j = 1, . . . , k, with the convention S0 := {�} for � the S-valued vectorof length 0. We alternatively call M an L-structure or simply a structure when the signature L

COMBINATORIAL LEVY PROCESSES 7

is understood. We write LS to denote the set of L-structures over S. We call i j the arity of M j foreach j = 1, . . . , k.

Remark 3.3 (Components with arity 0). We exclude the case i1 = · · · = ik = 0 from Definition3.2 for technical reasons. By the convention S0 = {�}, the space LS of structures with signatureL = (0) consists of the two elements M = (S; ∅) and M = (S; {�}). Therefore, although the casei1 = · · · = ik = 0 is not particularly interesting when k = 1, it is still a nontrivial state space onwhich to define a process. For k > 1, the structure M = (S; M1, . . . ,Mk) with signature (0, . . . , 0)corresponds to an element in the hypercube, which is of interest in various applications includingthe design of experiments.

Example 3.4 (Common examples). In terms of Definition 3.2, a subset A ⊆N is a combinatorialstructure with L = (1), that is, A ⊆ N corresponds to (N; A). A directed graph G with vertexsetN and edge set E ⊆ N ×N is a structure with L = (2), that is, G = (N; E). (Our definitionhere permits self-loops in G.) Taking L = (1, 2), we obtain M = (N; A,E), which correspondsto a graph (N; E) and a designated subset, or community, of vertices A ⊆ N. For L = (1, 2, 3),M = (N; A1,A2,A3) represents first-, second-, and third-order interactions among a collection ofparticles or among statistical units in a designed experiment.

The act of subsampling S′ ⊆ S induces a natural restriction operation LS → LS′ byM 7→M|S′ , where

(8) M|S′ := (S′; M1 ∩ S′i1 , . . . ,Mk ∩ S′ik).

Any permutation σ : S→ S induces a relabeling operation LS → LS by M 7→Mσ, where

(9) Mσ := (S; Mσ1 , . . . ,M

σk )

is defined by

(a1, . . . , ai j) ∈Mσj if and only if (σ(a1), . . . , σ(ai j)) ∈M j for each j = 1, . . . , k.

Combining (8) and (9), we define the image of M ∈ LS by any injection ϕ : S′ → S asMϕ = (S′; Mϕ

1 , . . . ,Mϕk ) ∈ LS′ , where

(10) (a1, . . . , ai j) ∈Mϕj if and only if (ϕ(a1), . . . , ϕ(ai j)) ∈M j for each j = 1, . . . , k.

Under these operations, the spaceLN of countable combinatorial structures comes furnishedwith the product discrete topology induced by the ultrametric

(11) d(M,M′) := 1/(1 + sup{n ∈N : M|[n] = M′|[n]}), M,M′ ∈ LN,

with the convention 1/∞ = 0. Under (11), (LN, d) is a compact, separable, and Polish metricspace, which we equip with the Borel σ-algebra.

3.1. Combinatorial increments. For any S ⊆N and M = (S; M1, . . . ,Mk) ∈ LS, we write

M j(a) = 1{a ∈M j} :={

1, a ∈M j,0, otherwise,

for each a = (a1, . . . , ai j) ∈ Si j , j = 1, . . . , k. We then define the increment between M and M′

in LS by M4M′ = 4(M,M′) := (S; ∆1, . . . ,∆k), where

(12) a ∈ ∆ j if and only if M j(a) ,M′j(a),

for each a = (a1, . . . , ai j) ∈ Si j , j = 1, . . . , k. For example, when L = (1), M4M′ is thesymmetric difference between subsets of N as in (1); when L = (2), M4M′ is the directed

8 HARRY CRANE

graph whose edges are the pairs (i, j) at which M and M′ differ; and so on. Importantly, theincrement between any two L-structures is also an L-structure with the same base set.

The spaces of L-structures we consider can be regarded as a group (LN,4), where thegroup action 4 is defined by the increment operation above. In particular, every M ∈ LSacts on LS by M′ 7→M4M′. Defined in this way, (LS,4) is a transitive, abelian group withidentity given by the empty structure 0LS := (S; ∅, . . . , ∅) and for which every element M ∈ LSis its own inverse. The group structure ofLS enriches the product discrete topology inducedby (11) and underlies several key properties of combinatorial Levy processes. Furthermore,LS is partially ordered and has minimum element 0LS under pointwise inclusion, that is,M ≤M′ if and only if M j(a) ≤M′j(a) for every a ∈ Si j , for all j = 1, . . . , k.

3.2. Exchangeability and combinatorial limits. de Finetti’s theorem, the Aldous–Hoovertheorem, and their relatives permit the study of exchangeable sequences, graphs, and {0, 1}-valued arrays by projecting into a continuous limit space, for example, the unit interval,the space of graph limits, and the space of hypergraph limits, respectively. More generally,exchangeable combinatorial L-structures permit an analogous representation in a space ofcombinatorial limits. The example in Section 2 shows that much of the structural behaviorof an exchangeable set-valued Levy process is determined by its projection into the unitinterval. Our main theorems extend this idea to characterize exchangeable combinatorialLevy processes through their induced behavior in the appropriate limit space.

As mentioned in Section 2, the combinatorial limit of M ∈ LN is not as simple as theprojection of A ⊆N to its limiting frequency π(A) as in (7). To see why, consider A,A′ ⊆Nand let M = (N; A,A′) be the associated (1, 1)-structure. Although M is just a pair of subsets,the individual frequencies π(A) and π(A′) are not sufficient to summarize the full structureof M: if we construct A by including each element i ∈ N independently with probabilityp ∈ (0, 1) and we define A′ = A, then π(A) = π(A′) = p with probability 1; but if we defineA and A′ as independent and identically distributed so that each element has probabilityp ∈ (0, 1) of appearing in A, respectively A′, then π(A) = π(A′) = p with probability 1, butP{A = A′} = 0. In both cases, (π(A), π(A′)) = (p, p), but the structure of M = (N; A,A′)is vastly different in the two constructions. The pair (π(A), π(A′)) does not capture allstructural features of (N; A,A′), motivating the following definition.

Definition 3.5 (Homomorphism density). For any signature L and finite subsets S′ ⊆ S ⊂N,we define the homomorphism density of A ∈ LS′ in M ∈ LS by

(13) δ(A; M) :=1|S|↓|S′|

∑ϕ:S′→S

1{Mϕ = A},

where the sum is over injections ϕ : S′ → S, |S| denotes the cardinality of S ⊆ N, and n↓m :=n(n − 1) · · · (n −m + 1) is the falling factorial function. For brevity, we refer to (13) as the densityof A in M. Intuitively, δ(A; M) is the probability that Mϕ = A for ϕ chosen uniformly at randomamong all injections S′ → S.

For fixed M ∈ LS, the density function δ(·; M) determines a probability measure on LS′

for every S′ ⊆ S. For M ∈ LN and A ∈ L[m], we define the limiting density of A in M by

(14) δ(A; M) := limn→∞

δ(A; M|[n]), if it exists.

Provided each of the limits δ(A; M), A ∈ L[n], exists, the collection of homomorphismdensities (δ(A; M), A ∈ L[n]) determines a probability measure on L[n] by the bounded

COMBINATORIAL LEVY PROCESSES 9

convergence theorem. If (14) exists for every A ∈⋃

n∈N L[n], then the family of distributionsdefined by (δ(A; M),A ∈ L[n]) for each n ∈N determines a unique probability measure onLN, which we denote by ‖M‖.

Definition 3.6 (Combinatorial limit). We define the combinatorial limit ‖M‖ of M ∈ LN as theunique probability measure µ on LN such that

(15) µ({A∗ ∈ LN : A∗|[m] = A}) = δ(A; M), A ∈ L[m], m ∈N,

provided the limit δ(A; M) exists for every A ∈⋃

m∈N L[m]. For brevity, we write

‖M‖(A) := ‖M‖({A∗ ∈ LN : A∗|[m] = A}) for each A ∈ L[m], m ∈N .

Lovasz and Szegedy [29] defined the concept of a graph limit in terms of the limitinghomomorphism densities of all finite subgraphs within a sequence of graphs. Definition 3.6extends the Lovasz–Szegedy notion to the more general setting of combinatorial structuresfrom Definition 3.2. The space of exchangeable, dissociated probability measures plays afundamental role in the study of exchangeable structures.

Definition 3.7 (Exchangeable and dissociated L-structures). For any S ⊆ N, a randomstructure M = (S; M1, . . . ,Mk) is exchangeable if Mσ =DM for all permutations σ : S→ S thatfix all but finitely many elements of S. We call M ∈ LS dissociated if M|T and M|T′ are independentwhenever T,T′ ⊆ S are disjoint.

When A ⊆N is a random subset, exchangeable and dissociated corresponds to indepen-dent and identically distributed, which explains why the projection π(A) into [0, 1] in (7) isenough to determine the combinatorial limit ‖M‖ of a (1)-structure M = (N; A); see Equation(7) and the discussion at the end of Section 2. For more complex structures, dissociationstill allows dependence between certain parts of the structure. In Proposition 6.2, we provethat the combinatorial limit of any exchangeable L-structure exists with probability 1.

Definition 3.8 (Combinatorial limit space). For any signature L, we write EL to denote thespace of exchangeable, dissociated probability measures on LN.

As every W ∈ EL is a probability measure on LN, we write W(A), A ∈ L[n], as shorthandfor

W(A) := W({M ∈ LN : M|[n] = A}), A ∈ L[n] .

We then define the distance between W,W′ ∈ EL by

(16) d(W,W′) =∑n∈N

2−n∑

A∈L[n]

|W(A) −W′(A)|.

We equip EL with the Borel σ-algebra induced by this metric. The Borel σ-algebra is thesmallest σ-algebra such that ‖ · ‖ : LN → EL ∪{∂} is measurable, where we define ‖M‖ = ∂whenever ‖M‖ does not exist.

4. Summary of main theorems

4.1. General combinatorial Levy processes. Recall the definition of the increment 4 :LS ×LS → LS in (12) and 0LS = (S; ∅, . . . , ∅).

Definition 4.1 (Combinatorial Levy process). For any signature L and S ⊆ N, we callX = (Xt, t ≥ 0) on LS a combinatorial Levy process if it has

• X0 = 0LS ,

10 HARRY CRANE

• stationary increments, that is, 4(Xt+s,Xs) =DXt for all s, t ≥ 0,• independent increments, that is, 4(Xt1 ,Xt0), . . . ,4(Xtk ,Xtk−1) are independent for all

0 ≤ t0 ≤ t1 ≤ · · · ≤ tk < ∞, and• cadlag sample paths, that is, t 7→ Xt is right continuous and has left limits under the

product discrete topology induced by (11).

Remark 4.2. The first condition above, X0 = 0LS , is akin to the condition X0 = 0 for R-valuedLevy processes. By the stationarity and independence of increments, there is no loss of generality inassuming X0 = 0LS . A combinatorial Levy process Xx with initial state X0 = x can be obtained fromX = (Xt, t ≥ 0) started at 0LS by putting Xx := (Xx

t , t ≥ 0) with Xxt = Xt 4 x for all t ≥ 0.

In discrete time, combinatorial Levy processes are analogous to random walks, and mostof their structural properties follow directly from Definition 4.1.

Definition 4.3 (Combinatorial random walk). A (combinatorial) random walk on LS withincrement distribution µ and initial state X0 is a discrete time process X = (Xm,m ≥ 0) with

(17) Xm =D 4(Xm−1,∆m), m ≥ 1,

where ∆1,∆2, . . . are i.i.d. from µ.

Theorem 4.4. Let X = (Xm, m ≥ 0) be a discrete time combinatorial Levy process on LS. Thenthere exists a unique probability measure µ on LS such that X =D X∗µ = (X∗m, m ≥ 0), where X∗µ is acombinatorial random walk on LS with initial state X0 = 0LS and increment distribution µ.

In continuous time, a combinatorial Levy process on LN must balance its behavior sothat its sample paths satisfy the cadlag requirement: since each L[n] is a finite state space,X[n] := (Xt|[n], t ≥ 0) can jump only finitely often in bounded time intervals. On the otherhand, since we have ruled out the case i1 = · · · = ik = 0, X = (Xt, t ≥ 0) evolves on anuncountable state space and is defined at an uncountable set of times; therefore, X canexperience infinitely many discontinuities in any bounded time interval. Condition (18) inTheorem 4.5 strikes the balance.

Theorem 4.5. Let X = (Xt, t ≥ 0) be a continuous time combinatorial Levy process on LN. Thenthere is a unique measure µ on LN satisfying

(18) µ({0LN}) = 0 and µ({M∗ ∈ LN : M∗|[n] , 0L[n]}) < ∞ for all n ∈N

such that the infinitesimal jump rates of X satisfy

(19) limt↓0

1tP{Xt ∈ d∆} = µ(d∆), ∆ ∈ LN \{0LN},

where convergence in (19) is understood in the sense of vague convergence of σ-finite measures.

The limit in (19) is well defined on account of the Feller property for combinatorial Levyprocesses, as we now discuss. The stationary and independent increments assumptionsimply that X is a time homogeneous Markov process with transition law determined bythe Markov semigroup Q = (Qt, t ≥ 0), where

(20) Qt g(M) := Eg(Xt 4M), t ≥ 0,

for all bounded, continuous functions g : LN → R and all M ∈ LN. We call Q a Fellersemigroup and say that X has the Feller property if

• limt↓0 Qt g(M) = g(M) for all M ∈ LN and

COMBINATORIAL LEVY PROCESSES 11

• M 7→ Qt g(M) is continuous for every t > 0,for all bounded, continuous g : LN → R.

Proposition 4.6. An LN-valued process X = (Xt, t ≥ 0) is a combinatorial Levy process if andonly if X[n] = (Xt|[n], t ≥ 0) is a combinatorial Levy process on L[n] for every n = 1, 2, . . ..

From Proposition 4.6, we deduce the Feller property for combinatorial Levy processes.

Corollary 4.7. Every combinatorial Levy process has the Feller property.

Definition 4.8 (σ-finite measures). A measure µ on LN is σ-finite if it satisfies (18).

Given a σ-finite measure µ, we construct X∗µ = (X∗t , t ≥ 0) from a Poisson point process∆∗ = {(t,∆∗t)} ⊆ [0,∞) × LN with intensity measure dt ⊗ µ. For each n ∈ N, we constructX∗[n]µ = (X∗[n]

t , t ≥ 0) on L[n] by putting X∗[n]0 = 0L[n] and

• X∗[n]t = X∗[n]

t− 4 ∆∗t |[n], if (t,∆∗t) ∈ ∆∗ and ∆∗t |[n] , 0L[n], and

• X∗[n]t = X∗[n]

t− otherwise,

where X∗[n]t− := lims↑t X∗[n]

s is the state of X∗[n]µ at the instant before time t. (Notice that by the

last condition we tacitly construct X∗[n]µ to be constant between atom times of ∆∗.)

Since we construct each X∗[n]µ from the same Poisson point process ∆∗, the collection

(X∗[n], n ∈ N) is mutually compatible, that is, X∗[n]µ |[m] := (X∗[n]

t |[m], t ≥ 0) = X∗[m]µ for every

m ≤ n, and, thus, determines a unique process X∗µ = (X∗t , t ≥ 0) on LN.

Definition 4.9 (Canonical Levy processes). We call X∗µ a µ-canonical Levy process.

Theorem 4.10. Let X be a combinatorial Levy process with rate measure µ as in (18). ThenX =D X∗µ, where X∗µ is a µ-canonical Levy process. Conversely, every combinatorial Levy processX has the same finite-dimensional distributions as some canonical Levy process corresponding to aσ-finite measure µ.

4.2. Exchangeable processes.

Definition 4.11 (Exchangeable Levy process). An LS-valued process X = (Xt, t ≥ 0) isexchangeable if X =D Xσ for all permutations σ : S→ S that fix all but finitely many elements ofS.

The special case of graph-valued Levy processes relates to recent work on the theory ofgraph limits [29] and dynamic random networks [16]. Definition 3.6 extends the notionof graph limit to that of a combinatorial limit for general L-structures. By projectinginto the appropriate combinatorial limit space, the preceding theorems specialize nicelyto the exchangeable setting. Recall that the limit space EL consists of exchangeable,dissociated probability measures on LN. Given a measure ν on EL, we write ν∗ to denotethe exchangeable measure it induces on LN by

(21) ν∗(S) :=∫EL

W(S)ν(dW), S ⊆ LN .

As long as ν is a probability measure on EL, ν∗ is a probability measure on LN, but thedefinition in (21) is well defined for arbitrary positive measures ν.

For any combinatorial Levy process X = (Xt, t ≥ 0), we write ‖X ‖ = (‖Xt‖, t ≥ 0) todenote its projection into EL, if it exists. The next theorem says that ‖X ‖ always exists forexchangeable combinatorial Levy processes.

12 HARRY CRANE

Theorem 4.12. Let X = (Xm, m ≥ 0) be an exchangeable combinatorial Levy process in discrete time.Then there exists a unique probability measure ν onEL such that the increments of X are independentand identically distributed according to ν∗. Moreover, the projection ‖X ‖ = (‖Xm‖, m ≥ 0) existsalmost surely and is a Markov chain.

4.3. Levy–Ito structure. The final theorems explicitly characterize the measure µ guaran-teed by Theorem 4.10 for continuous time processes. A formal statement requires morenotation.

We often deal with unordered multisets, for which we also write {x1, . . . , xn} with theunderstanding that x1, . . . , xn need not be distinct. We can also express any multisetx = {x1, . . . , xn} by {imi : i ≥ 1}, where mi = |{ j ∈ [n] : x j = i}| is the multiplicity of elementi in x. For example, x = {1, 1, 2, 2, 2} has m1 = 2, m2 = 3, and m j = 0 for j ≥ 3 so thatx ≡ {12, 23

}, omitting elements with multiplicity 0 for convenience. Given two multisetsx, x′ with multiplicities m = (m1,m2, . . .) and m′ = (m′1,m

′

2, . . .), respectively, we write x � x′

to denote that mi ≤ m′i for all i ≥ 1 and we define the intersection x ∩ x′ to be the multisetwith multiplicities mi ∧m′i for each i ≥ 1. When necessary, we write [x] := { j ≥ 1 : m j > 0}to denote the set of elements in x without multiplicity. We apply the same notation forordered multisets x = (x1, . . . , xn) when the order of elements is inconsequential, as in theconditions of (22) below.

Let L = (i1, . . . , ik) be a signature and s = {s1, . . . , sq} ⊂ N be a multiset, for someq = 0, 1, . . . , ik. For any M ∈ LN, we define an L-structure M∗s = (N; M∗s,1, . . . ,M

∗

s,k) by

(22) M∗s, j(a) =

M j(a), |a| ≤ |s|, a � s, [a] = [s],M j(a), |a| > |s|, s � a, [a] ⊇ [s],

0, otherwise,a ∈Ni j , j = 1, . . . , k.

Therefore, M∗s is theL-structure that corresponds to M on supersets of s and to 0LN

otherwise.We call M∗s the s-substructure of M.

Remark 4.13. The two separate conditions in (22) are needed to fully capture all possible behaviorsin our main theorem below. The subset s = {s1, . . . , sq} in (22) represents the elements indexing thechosen substructure of M. If |s| ≥ i j for some component j = 1, . . . , k of the signatureL = (i1, . . . , ik),then M∗s, j(a) is nonzero only if a is a proper subset of s in the sense that the multiplicities of a are nogreater than s and all elements in a are also in s. If |s| < i j, then M∗s is nonzero only if all elementsof s appear in a with multiplicity at least their multiplicity in s. Some examples should clarify thisdefinition.

Let L = (1, 2) so that M = (N; A,E) is a set A ⊆ N together with a graph (N; E). Fors = {1}, M∗s retains only relations in M involving element 1. Specifically, M∗s, j(a) = 0 for alltuples a except possibly those containing element 1:

M∗s,1(( j)) =

{M1((1)), j = 1,

0, otherwise, and

M∗s,2(( j, j′)) =

{M2(( j, j′)), j = 1 or j′ = 1,

0, otherwise.

With regard to (22), we have |s| = 1 so that M∗s,1 is determined by the top line of (22), withthe only nontrivial contribution from a = (1), and M∗s,2 is determined by the second lineof (22), with nontrivial contributions from all a such that [a] ⊇ {1}. We note the difference

COMBINATORIAL LEVY PROCESSES 13

when s = {1, 1}, which has [s] = {1} but should not be confused with the singleton {1} in thecontext of (22). In this case,

M∗s,1(( j)) =

{M1((1)), j = 1,

0, otherwise, and

M∗s,2(( j, j′)) =

{M2((1, 1)), ( j, j′) = (1, 1),

0, otherwise.

Once again, M∗s,1 is determined by the top line of (22), for which the only nontrivialcontribution must have [a] = {1} and, therefore, a = (1). In contrast to the case s = {1},however, the top line of (22) also applies to M∗s,2, since we now have |a| ≤ |s|. The contributionM∗s,2(a) is nontrivial only if [a] = [s] = {1}, that is, a = (1, 1). Therefore, M∗

{1} and M∗{1,1} are

different structures in general. As a special case, we point out that M∗∅

= M for all M ∈ LN.Any multiset s = {s1, . . . , sq} ⊂ N determines a partition of the integer q, written λ(s) =

1λ12λ2 · · · qλq , where

λ j := |{r ∈ s : |{i ∈ [q] : si = r}| = j}|, j = 1, . . . , q,

is the number of elements that appear with multiplicity j in s. In general we write λ ` q toindicate that λ = 1λ12λ2 · · · qλq is a partition of the integer q, which must satisfy λi ≥ 0 for alli = 1, . . . , q and

∑qi=1 iλi = q.

For j = 1, . . . , k and s = {s1, . . . , sq}, q = 0, 1, . . . , i j, we can express each component M∗s, j of

M∗s as a structure with signature (i j − q)k j = (i j − q, . . . , i j − q) with k j equal arities, where

k j :=(

i j

i j − q

)q!∏q

l=1 l!λl

for λ(s) = 1λ12λ2 · · · qλq . (Note that k j is the number of all possible ways to insert theelements of s in an i j-tuple in any possible order.) For example, consider the case of i j = 3and s = {1, 2}, so that q = 2, λ(s) = 1220, and k j = 6 corresponds to the six tuples of the form

(∗, 1, 2), (∗, 2, 1), (1, ∗, 2), (2, ∗, 1), (1, 2, ∗), (2, 1, ∗),

where entries ∗ can be filled with arbitrary indices. In this case, we express M∗s, j =

(N; M∗s, j,1, . . . ,M∗

s, j,6), where each M∗s, j,l ⊆ Ni j−q = N. With the indices l = 1, . . . , 6 corre-

sponding to the ordering of tuples above, we have, for example,

M∗s, j,1((a)) = M j((a, 1, 2)),

M∗s, j,2((a)) = M j((a, 2, 1)),

M∗s, j,3((a)) = M j((1, a, 2)),

and so on. For s ⊂N with λ(s) = λ, we write Lλ to denote the signature of M∗s.For every s ⊂N, we define ‖ · ‖s by

(23) ‖M‖s = (‖M∗s,1‖, . . . , ‖M∗

s,k‖),

where ‖M∗s, j‖ is the combinatorial limit of M∗s, j as an (i j−q)k j-structure, with any prespecifiedconvention for ordering the components of M∗s, j = (M∗s,1, . . . ,M

∗

s,k j). We write ‖M‖s = 0 if

and only if ‖M∗s, j‖ = 0(i j−q)kj for all j = 1, . . . , k, where recall 0L is the combinatorial limit of

the empty structure 0LN

with signature L.

14 HARRY CRANE

For any λ = 1λ1 · · · qλq ` q, we define πλ = (π1, . . . , πp) by π1 ≥ π2 ≥ · · · ≥ πp > 0 such thatπ1 + · · · + πp = q and

|{k ∈N : |{i ∈ [q] : πi = k}| = j}| = λ j for each j = 1, . . . , q.

We define the canonical λ-multiset by sλ = {1π1 , 2π2 , . . . , pπp} so that each i appears πi timesin sλ. For example, if λ = 12213041 then πλ = (4, 2, 1, 1) and sλ = {1, 1, 1, 1, 2, 2, 3, 4}.

For any s ⊂Nwithλ(s) = λ, we index s = {sπ11 , . . . , s

πpp } so that si < si+1 wheneverπi = πi+1

and we define the canonical mapping σs,λ : [p]→ s by σs,λ(i) = si for each i = 1, . . . , p. Forexample, let s = {1, 1, 3, 4, 4, 5, 5, 5, 5} so that λ(s) = 11223041, sλ = {1, 1, 1, 1, 2, 2, 3, 3, 4}, andπλ = (4, 2, 2, 1) . Then we write s = {54, 12, 42, 31

} and σs,λ : [4] → s assigns σs,λ(1) = 5,σs,λ(2) = 1, σs,λ(3) = 4, and σs,λ(4) = 3, so that sσs,λ = sλ.

The above preparation anticipates Theorem 4.14 in which we decompose exchangeableσ-finite measures on LN according to how they handle various substructures. Below wewrite µλ to denote a measure on LN that satisfies (18),

(24) is invariant with respect to permutations that fix sλ,

(25) M∗sλ = M for µλ-almost every M ∈ LN,

and

(26)⋂{s′ : ‖M‖s′ , 0} = sλ for µλ-almost every M ∈ LN,

where the intersection of multisets is defined at the beginning of Section 4.3.We then define

(27) µ∗λ(·) =∑

s⊂N:λ(s)=λ

µλ({M ∈ LN : Mσ−1s,λ ∈ ·}).

For example, let λ = 11 be the only partition of integer 1 and

µλ((N; {i})) =

{c, i = 1,0, otherwise,

for some c > 0. Then µλ satisfies (18), (24), (25), and (26). For any k > 1, we note that s = {k}has λ = 11, πλ = (1), and σs,λ(1) = k so that Mσ−1

s,λ = (N; {1}) if and only if M = (N; {k}). Inthis case, µ∗λ assigns mass c to each singleton subset (N; {k}), k ∈ N, so µ∗λ is exchangeableand satisfies (18). Compare the definition of µ∗

11 to that of c∑∞

i=1 εi in Theorem 2.6.

Theorem 4.14 (Levy–Ito–Khintchine representation for combinatorial Levy processes). LetL = (i1, . . . , ik) be any signature and X = (Xt, t ≥ 0) be an exchangeable combinatorial Levy processon LN. Then there exists a unique measure ν0 on EL satisfying

(28) ν0({0L}) = 0 and∫EL

(1 −W({0L[ik]}))ν0(dW) < ∞,

and measures µλ on LN satisfying (18), (24), (25), and (26) such that

(29) µ = ν∗0 +∑

q=1,...,ik

∑λ`q

µ∗λ,

where λ ` q denotes that λ is a partition of q and µ∗λ is defined in (27).

COMBINATORIAL LEVY PROCESSES 15

We call (29) the Levy–Ito–Khintchine representation for exchangeable combinatorial Levyprocesses. In a precise sense, see Theorem 4.15, ν∗0 describes the discrete component of Xand the µλ decompose the continuous component of X.

Theorem 4.15. Let X = (Xt, t ≥ 0) be an exchangeable Levy process on LN. Then the projection‖X ‖ = (‖Xt‖, t ≥ 0) into EL exists almost surely and is a Feller process. Moreover, the samplepaths of ‖X ‖ are continuous except at the times of jumps from the ν∗0 measure in (29).

Much of our remaining effort is dedicated to proving Theorems 4.14 and 4.15. Weorganize the next few sections as follows. We first illustrate the above theorems in specific,concrete cases. We then discuss combinatorial limits and prove a precursor to Theorem4.14 before deducing the main theorems.

5. Examples

We couch the above theorems in terms of some specific combinatorial Levy processes,beginning with a summary of the set-valued Levy processes from Section 2 and then movingon to graph-valued processes. Finally, we combine set- and graph-valued processes todemonstrate how higher order structures evolve according to (29).

5.1. Set-valued Levy processes. In Section 2, we discussed combinatorial Levy processesin the special case when L = (1) and X = (Xt, t ≥ 0) evolves on the space of subsets ofN.In this case, the combinatorial limit of (N; A) is determined by the limiting frequency ofelements in a subset A ⊆N,

π(A) = limn→∞

n−1|A ∩ [n]|.

de Finetti’s theorem implies that the marginal distribution of X at any fixed time t ≥ 0 isdetermined by a unique probability measure ν on [0, 1] as in (4).

In the context of Theorem 4.14, the behavior of X on LN is described by a measureµ = ν∗ + c

∑∞

i=1 εi with components defined as in Theorem 2.6. The first component ν∗ isinduced from a measure ν satisfying (5), the analog to (28) in the special case of set-valuedprocesses. The second component c

∑∞

i=1 εi plays the role of µ∗11 in (29) since λ = 11 is the

only partition of the integer 1. The only nontrivial measures on the (0)-structure that satisfy(18), (24), (25), and (26) must be of the form µ11({∅}) = 0 and µ11({1}) = c ≥ 0. Our definitionof µ∗λ in (27) gives µ∗

11(·) = c∑∞

i=1 εi(·). The contribution of µ∗11 to the characteristic measure

of X is as discussed previously: each i ∈ N changes status independently at rate c ≥ 0,while the rest of X remains constant.

5.2. Graph-valued Levy processes. Let X = (Xt, t ≥ 0) be a Levy process on the space ofdirected graphs, possibly with self-loops, so that X evolves on the space of L-structureswith L = (2). By Theorem 4.14, the first component of µ in (29) is a measure ν0 on thespace of graph limits satisfying (28). The second component is decomposed according tothe three partitions 11, 1220, and 1021 of the integers 1 and 2 as follows.

(11) µ11 is a measure onLN for which almost every M = (N,E) has M = M∗{1} and at least

one of the conditions

limn→∞

n−1n∑

j=1

1{(1, j) ∈ E} > 0 or limn→∞

n−1n∑

j=1

1{( j, 1) ∈ E} > 0

holds.

16 HARRY CRANE

(1220) µ1220 assigns 0 mass to all M = (N; E) except that for which at least one of (1, 2) ∈ Eand (2, 1) ∈ E holds and (i, j) < E otherwise.

(1021) µ1021 assigns 0 mass to all M = (N; E) except that for which (1, 1) ∈ E and (i, j) < Eotherwise.

The jump rates of X are determined by µ = ν∗ + µ∗11 + µ∗

1220 + µ∗1021 . At the time of a

discontinuity in X, either

(0) a strictly positive proportion of edges changes status according to a σ-finite measureν∗0 on countable graphs,

(11) a positive proportion of edges incident to a specific vertex changes status and otheredges stay fixed,

(1220) edges involving a specific pair {i, j}, i , j, change status and the rest of the graphstays fixed, or

(1021) a single self-loop (i, i) changes status for a specific i ∈ N and the rest of the graphstays fixed.

In this special case, the limit process ‖X ‖ = (‖Xt‖, t ≥ 0) evolves on the space of graphlimits. Lovasz and Szegedy [29] introduced the term graph limit in 2006, but a more generalconcept originates with the Aldous–Hoover theorem in the late 1970s; see [5, Theorem14.11].

5.3. Networks with a distinguished community. Combining the structures in the previoustwo sections, we get signature L = (1, 2), which corresponds to a structure M = (N; A,E)with A ⊆N and E ⊆N ×N. In this case, a combinatorial Levy process X = (Xt, t ≥ 0) offersthe interpretation as the evolution of a network along with a distinguished community ofits vertices. As in the previous section, we must consider partitions of integers 1 and 2, soTheorem 4.14 characterizes exchangeable processes X by a σ-finite measure ν0 on EL andmeasures µ11 , µ1220 , µ1021 . The ν0 measure governs a joint evolution of the community andthe network such that atoms from ν0 cause a positive proportion of elements to changecommunity status and/or a positive proportion of edges to change status. The µλ measuresplay a similar role to Section 5.2 with some modifications. For 11, µ11 allows for the statusof element 1 to change in the subset A as well as a change to a positive proportion of edgesincident to element 1 as in Section 5.2. For 1220, µ1220 is just as in Section 5.2: there is achange to at least one of the edges (1, 2) and (2, 1) and no change in the community structureA. For 1021, µ1021 allows for a change to the status of element 1 in the community structureas well as a change to the status of edge (1, 1) in E.

6. Characterization of exchangeable σ-finite measures

6.1. Limits of combinatorial structures. Recall definition (14) of the limiting densities ofa structure M.

Theorem 6.1 (Aldous–Hoover theorem for L-structures [4, 24]). Let L = (i1, . . . , ik) bea signature and M be an exchangeable L-structure over N. Then there exists a measurablefunction g = (g1, . . . , gk) with g j : [0, 1]2i j

→ {0, 1} for each j = 1, . . . , k such that M =DMg =

(N; Mg1 , . . . ,M

gk ), where

Mgj (a) = g j((ξs)s⊆a), a = (a1, . . . , ai j) ∈N

i j ,

COMBINATORIAL LEVY PROCESSES 17

for (ξs)s⊂N:|s|≤ik a collection of i.i.d. Uniform[0, 1] random variables. In particular, M is conditionallydissociated given its tail σ-field.

Proposition 6.2. Let M = (N; M1, . . . ,Mk) be an exchangeable L-structure. Then δ(A; M) existsalmost surely for every A ∈ L[m], for all m = 1, 2, . . .. Moreover, the collection (δ(A; M), A ∈⋃

m∈N L[m]) exists almost surely and determines a unique probability measure ‖M‖ on LN.

Proof. In addition to being exchangeable, we first assume that M is dissociated, that is,M|S and M|T are independent whenever S and T are disjoint. For a fixed L-structureA = ([m]; A1, . . . ,Ak) over [m], we define

Zn :=1

n↓m∑

ϕ:[m]→[n]

1{M|ϕ[n] = A}, for each n = 1, 2, . . . .

Under uniform selection of an injection ϕ : [m] → [n], the σ-field Fn := σ〈Zn+1,Zn+2, . . .〉induces

P{M|ϕ[n] = A | Fn} = Zn+1, for each n = 1, 2, . . . .

Thus,

E(Zn | Fn) = E

1n↓m

∑ϕ:[m]→[n]

1{M|ϕ[n] = A} | Fn

= Zn+1

and (Zn, n ∈ N) is a reverse martingale. By the reverse martingale convergence theorem,there exists a random variable Z∞ such that Zn → Z∞ almost surely. Since we haveassumed M is dissociated, the limit depends only on the tail σ-field T =

⋂n∈N Fn and, thus,

is deterministic by the 0-1 law. That δ(A; M) exists for any exchangeable M follows by thefact that any exchangeable L-structure is conditionally dissociated given its tail σ-field,by Theorem 6.1. Almost sure existence of the infinite collection (δ(A; M), A ∈

⋃m∈N L[m])

follows by countable additivity of probability measures.To prove that (δ(A; M), A ∈

⋃m∈N L[m]) determines a unique, exchangeable probability

measure on LN, we consider A ∈ L[m] and A′ ∈ L[n] such that A′|[m] = A, for m ≤ n. Forfixed r ≥ n, the definition in (13) implies∑

A′∈L[n]: A′|[m]=A

δ(A′; M|[r]) =∑

A′∈L[n]: A′|[m]=A

1r↓n

∑ϕ:[n]→[r]

1{M|ϕ[r] = A′}

=1

r↓n∑

ϕ:[n]→[r]

∑A′∈L[n]: A′|[m]=A

1{M|ϕ[r] = A′}

=1

r↓n∑

ϕ:[m]→[r]

1{M|ϕ[r] = A}∑

extensions of ϕ to [n]→[r]

1

=1

r↓n∑

ϕ:[m]→[r]

1{M|ϕ[r] = A} × (r −m)(r −m − 1) · · · (r − n + 1)

=1

r↓m∑

ϕ:[m]→[r]

1{M|ϕ[r] = A}

= δ(A; M|[r]).

Since r ≥ n is arbitrary, the probability measures induced onL[m] andL[n] are consistent forall m ≤ n. Caratheodory’s extension theorem implies an extension to a unique probability

18 HARRY CRANE

measure on LN. Since each of the finite space distributions is exchangeable, so is thedistribution induced on LN. �

By Proposition 6.2, every exchangeable L-structure projects to a unique limit in EL.Conversely, the law of every exchangeable L-structure M is determined by a probabilitymeasure ν on EL such that M ∼ ν∗, where ν∗ is defined in (21). By the projective structureof LN, ν∗ is uniquely determined by the induced measures

ν∗(n)(M) := ν∗({M∗ ∈ LN : M∗|[n] = M}), M ∈ L[n],

for every n ∈N.

6.2. σ-finite measures. We are especially interested in Levy processes that evolve in con-tinuous time and, therefore, can jump infinitely often in bounded time intervals. To seethe additional possibilities in this case, let L = (1) so that µ is an exchangeable measure onsubsets ofN. For c > 0, we define

µ(dM) = c∞∑

i=1

1{M = (N; {i})},

which assigns mass c to the singleton subsets ofN and, thus, has infinite total mass. Forn = 1, 2, . . ., the restriction of µ to L[n] is

µ(n)(M) =

{c, M = ([n]; {i}),∞, otherwise,

which is finite and exchangeable onL[n] \{([n]; ∅)}. On the other hand, let c, c′ ≥ 0 and define

µ(dM) = c∞∑

i=1

1{M = (N; {i})} + c′∞∑

i=1

∞∑j=i+1

1{M = (N; {i, j})},

so that singletons have mass c and doubletons have mass c′. For n ∈N,

µ(n)(M) = cn∑

i=1

1{M = ([n]; {i})} + c′n∑

i=1

∞∑j=n+1

1{M = ([n]; {i})} + c′n−1∑i=1

n∑j=i+1

1{M = ([n]; {i, j})},

which is finite only if c′ = 0. (The middle term in the above expression results because therestriction of any (N; {i,n + j}) to [n] is ([n]; {i}), for every j = n + 1,n + 2, . . ..) Immediately,µ satisfies (18) only if it assigns no mass to doubleton subsets. The same argument rulesout tripletons, quadrupletons, and so on.

Theorem 6.3. Let L = (i1, . . . , ik) be a signature and µ be an exchangeable measure on LN thatsatisfies (18). Then there exists a unique measure ν0 on EL satisfying (28) and measures µλsatisfying (18), (24), (25), and (26) such that

(30) µ = ν∗0 +

ik∑q=1

∑λ`q

µ∗λ,

for µ∗λ defined in (27).

We first show that any µ constructed as in (30) satisfies (18).

Proposition 6.4. Let ν0 satisfy (28). Then ν∗0 in (21) satisfies (18).

COMBINATORIAL LEVY PROCESSES 19

Proof. The lefthand side of (18) follows immediately from the lefthand side of (28). For therighthand side of (18), we need to show

ν∗0({M ∈ LN : M|[n] , 0L[n]}) < ∞ for all n ∈N .

We note that

{M ∈ LN : M|[n] , 0L[n]} =⋃

s={s1<···<sik }⊂[n]

{M ∈ LN : M|s , 0Ls },

because M|[n] = 0L[n] only if M|s is trivial for all s ⊂ [n] with |s| = ik. By exchangeability of ν∗0,

ν∗0({M ∈ LN : M|s , 0Ls }) =

∫EL

(1 −W({0L[ik]}))ν0(dW)

for every s = {s1 < · · · < sik} ⊆ [n]. Thus,

ν∗0({M ∈ LN : M|[n] , 0L[n]}) = ν∗0

⋃s={s1<···<sik }⊂[n]

{M ∈ LN : M|s , 0Ls }

≤

∑s={s1<···<sik }⊂[n]

ν∗0({M ∈ LN : M|s , 0Ls })

≤ nik

∫EL

(1 −W({0L[ik]}))ν0(dW)

< ∞,

by the righthand side of (28). The proof is complete. �

Proposition 6.5. Let L = (i1, . . . , ik) be a signature, q = 1, . . . , ik, λ ` q, and suppose that µλ is ameasure on LN satisfying (18), (24), (25), and (26). Then µ∗λ, as defined in (27), satisfies (18).

Proof. Let λ ` q for some q = 1, . . . , ik. By (25), M = M∗sλ for µλ-almost every M ∈ LN;

whence, µλ({M ∈ LN : Mσ−1s,λ |[n] , 0L[n]}) = 0 for s = {s1, . . . , sq} * [n]. For s = {s1, . . . , sq} ⊂ [n],

we observe that Mσ−1s,λ |[n] = M|

σ−1s,λ

[n] and, therefore,

µλ({M ∈ LN : Mσ−1s,λ |[n] , 0L[n]}) = µλ({M ∈ LN : M|

σ−1s,λ

[n] , 0L[n]}) = µλ({M ∈ LN : M|[n] , 0L[n]}).

It follows that

µ∗λ({M ∈ LN : M|[n] , 0L[n]}) =∑

s⊂N:λ(s)=λ

µλ({M ∈ LN : Mσ−1s,λ |[n] , 0L[n]})

=∑

s⊂[n]:λ(s)=λ

µλ({M ∈ LN : M|σ−1

s,λ[n] , 0L[n]})

=∑

s⊂[n]:λ(s)=λ

µλ({M ∈ LN : M|[n] , 0L[n]})

≤ nqµλ({M ∈ LN : M|[n] , 0L[n]})< ∞,

20 HARRY CRANE

which establishes the righthand side of (28). The lefthand side of (28) follows from assumingthat µλ satisfies (18), the fact that there are at most countably many s = {s1, . . . , sq} for whichλ(s) = λ, and countable additivity of measures. �

Theorem 6.3 states the converse: the above construction is true of every exchangeableσ-finite measure on LN. We prove Theorem 6.3 in several steps.

Lemma 6.6. Let µ be an exchangeable σ-finite measure on LN. Then ‖M‖s exists for all s ={s1, . . . , sq} ⊂N, q = 0, 1, . . . , ik, for µ-almost every M ∈ LN.

Proof. Recall Definition 3.6 of combinatorial limit. Fix n = 1, 2, . . . and define µn as therestriction of µ to the event {M ∈ LN : M|[n] , 0L[n]}. By the righthand side of (18), µn isfinite, because

µn(S) = µ({M ∈ LN : M ∈ S and M|[n] , 0L[n]}) ≤ µ({M ∈ LN : M|[n] , 0L[n]}) < ∞

for all measurable sets S ⊆ LN, by the righthand side of (18). Furthermore, µn is invariantwith respect to permutations of N that fix [n]. We define the shifted measure ←−µ n as theimage of µn by M 7→

←−Mn := (N;

←−Mn

1 , . . . ,←−Mn

k ), where

(31) (a1, . . . , ai j) ∈←−Mn

j if and only if (a1 + n, . . . , ai j + n) ∈M j,

for each j = 1, . . . , k. We call←−Mn the n-shift of M. For example, if M = (N; {1, 2, 5, 6, 8}), then

←−M1 = (N; {1, 4, 5, 7, }),

←−M2 = (N; {3, 4, 6}), and so on.

The n-shifted measure ←−µ n is exchangeable and finite; therefore, ←−µ n is proportional toan exchangeable probability measure and Proposition 6.2 implies that ←−µ n-almost everyM ∈ LN possesses a unique limit ‖M‖. Furthermore,←−µ n induces a unique finite measure‖←−µ n‖ on EL by

‖←−µ n‖(dW) :=←−µ n({M ∈ LN : ‖M‖ ∈ dW}).

Since ‖M‖ depends only on the n-shift←−Mn, for every n ∈ N, µn-almost every M ∈ LN

possesses a limit as well. Finally, notice that the events {M ∈ LN : M|[n] , 0L[n]} increase to

{M ∈ LN : M , 0LN} as n→∞. Since we have assumed that µ assigns no mass to {0L

N}, the

monotone convergence theorem implies that µn ↑ µ as n → ∞, and thus µ-almost everyM ∈ LN possesses a limit ‖M‖.

The above argument shows that ‖M‖s exists for µ-almost every M when s = ∅. Theargument is similar for s = {s1, . . . , sq} ⊂ N with q = 1, . . . , ik. We define µn,s to be themeasure induced by µn through the map M 7→ M∗s. For any measurable set S ⊆ LN, wedefine S∗s := {M∗s : M ∈ S}, and we see that

µn,s(S∗s) = µ({M ∈ LN : M∗s ∈ S∗s and M|[n] , 0L[n]}) ≤ µ({M ∈ LN : M|[n] , 0L[n]}) < ∞.

Furthermore, µn,s is invariant with respect to permutations that fix [n] and s. Takingn ≥ 1 + max1≤ j≤q s j, we may define the n-shift measure←−µ n,s just as before, so that←−µ n,s isexchangeable and finite for every s ⊆ [n]. It follows that ‖M‖s exists for←−µ n,s-almost everyM ∈ LN. Since ‖M‖s depends only on M∗s, it follows that ‖M‖s exists for←−µ n-almost everyM ∈ LN and, by monotone convergence, µ-almost every M ∈ LN. �

When µ is a probability measure, the meaning of the limit ‖M‖ for M ∼ µ is straightfor-ward, for example, ‖M‖ = 0L indicates that M = 0L

Nalmost surely. However, our running

example for subsets shows why this is not the case for σ-finite measures µ; see the discussion

COMBINATORIAL LEVY PROCESSES 21

in Section 5.1. In particular, µ can assign positive mass to singleton subsets M = (N; {i}),for every i = 1, 2, . . ., for which ‖M‖ = 0L but M , (N; ∅) = 0L

N. To characterize µ, we must

understand how it treats events of the form {M ∈ LN : ‖M‖ = 0L}.

Lemma 6.7. Suppose µ is exchangeable, σ-finite, and µ-almost every M ∈ LN has ‖M‖ , 0L .Then there exists a unique measure ν on EL satisfying (28) such that µ = ν∗.

Proof. As in Lemma 6.6, we let µn denote the restriction of µ to {M ∈ LN : M|[n] , 0L[n]} andwe write←−µ n as the image of µn by the n-shift operation (31). Since←−µ n is exchangeable, thecombinatorial limit ‖M‖ exists for←−µ n-almost every M ∈ LN, allowing us to write

←−µ n(dM) =

∫EL

W(dM)‖←−µ n‖(dW).

By assumption, µn-almost every M has ‖M‖ , 0L , from which it follows that

µn({←−Mn|[ik] , 0L[ik]}) =

∫EL

(1 −W({0L[ik]}))‖←−µ n‖(dW),

where←−Mn is the n-shift from (31). Again, µn ↑ µ implies ‖←−µ n‖ ↑ ν for some measure ν;

whence, ν({0L}) = 0 and

µn({←−Mn|[ik] , 0L[ik]}) ↑

∫EL

(1 −W({0L[ik]}))ν(dW).

Furthermore, µn ↑ µ implies

µn({←−Mn|[ik] , 0L[ik]}) ≤ µ({

←−Mn|[ik] , 0L[ik]}) = µ({M|[ik] , 0L[ik]}) < ∞,

by the righthand side of (18) and exchangeability. It follows that∫EL

(1 −W({0L[ik]}))ν(dW) < ∞,

so that ν satisfies (28).To establish the identity µ = ν∗, we observe that

µ({M∗ ∈ LN : M∗|[n] = M, ‖M∗‖ , 0L}) =

= limm↑∞

µ({M∗ ∈ LN : M∗|[n] = M, ‖M∗‖ , 0L ,←−M∗n|[m] , 0L[m]}),

for every fixed n ∈ N and M ∈ L[n] \{0L[n]}. The identity follows by the monotone conver-gence theorem, because

{M∗ ∈ LN : M∗|[n] = M, ‖M∗‖ , 0L ,←−M∗n|[m] , 0L[m]}

increases to

{M∗ ∈ LN : M∗|[n] = M, ‖M∗‖ , 0L ,←−M∗n , 0L

N} as m→∞,

and {M∗ ∈ LN : ‖M∗‖ , 0L} ⊆ {M∗ ∈ LN :←−M∗n , 0L

N} for every n ∈N. Exchangeability of µ

allows us to permute the blocks {1, . . . ,n} and {n + 1, . . . ,n + m} so that

µ({M∗ ∈ LN : M∗|[n] = M,←−M∗n|[m] , 0L[m], ‖M

∗‖ , 0L}) =←−µm({M∗ ∈ LN : M∗|[n] = M, ‖M∗‖ , 0L}).

22 HARRY CRANE

Now,←−µm is exchangeable and previous arguments imply

←−µm({M∗ ∈ LN : M∗|[n] = M, ‖M∗‖ , 0L}) =

∫EL

W({M∗|[n] = M})‖←−µm‖(dW),

which converges to ∫EL

W({M∗|[n] = M})ν(dW) = ν∗({M∗|[n] = M}).

Since we chose n arbitrarily and we restricted ‖M∗‖ , 0L so that M∗ , 0LN

, we must haveequality of µ and ν∗ on the π-system that generates the Borel σ-field. Since σ-finite measuresare determined by their behavior on a generating π-system, the proof is complete. �

Lemma 6.8. Let µ be an exchangeable, σ-finite measure on LN for which µ-almost every M ∈ LNhas ‖M‖ = 0L . Then for each λ ` q, q = 1, . . . , ik, there are unique measures µλ satisfying (24),(25), and (26) such that

µ =

ik∑q=1

∑λ`q

µ∗λ.

Proof. For M ∈ LN, we define s(M) = sM :=⋂{s′ ⊂ N : ‖M‖s′ , 0}, provided ‖M‖s′

exists for all multisets s′ ⊂ N with |s′| ≤ ik. We then define λ(M) = λM := λ(sM), whereλ(s) = 1λ12λ2 · · · qλq is the partition induced by sM. By Lemma 6.6, ‖M‖s′ exists for all s′ ⊂Nfor µ-almost every M ∈ LN and, therefore, sM and λM are well defined for µ-almost everyM ∈ LN. For every λ ` q, q = 1, . . . , ik, we define %∗λ as the restriction of µ to the event{M ∈ LN : λM = λ}, that is, %∗λ = µ1{M∈LN:λM=λ}. It follows that µ =

∑ikq=1

∑λ`q %

∗

λ since the %∗λare mutually singular for different λ. Each ρ∗λ inherits σ-finiteness and exchangeability fromµ. For λ ` q, we define µλ = µ1{M∈LN:sM=sλ}, which satisfies (25) by exchangeability and (26)by definition. Also by exchangeability, we must have %∗λ = µ∗λ. The proof is complete. �

Proof of Theorem 6.3. This is a consequence of Proposition 6.4 and Lemmas 6.6, 6.7, and6.8. �

7. Proofs of main theorems

Theorem 6.3 is the key to our main conclusions about combinatorial Levy processes. Inthis section, we prove the main theorems from Section 4.

7.1. Discrete-time combinatorial Levy processes. Theorem 4.4 is immediate from Defini-tion 4.1. We now prove Theorem 4.12, which deals with discrete time combinatorial Levyprocesses that are exchangeable.

Proof of Theorem 4.12. Let X = (Xm, m ≥ 0) be an exchangeable combinatorial Levy process indiscrete time. By definition, the increments process (∆m, m ≥ 1) defined by ∆m := Xm 4Xm−1is a sequence of independent, identically distributed structures. The increment operator 4satisfies

∆σm = (Xm 4Xm−1)σ = Xσm 4Xσ

m−1.

By exchangeability of X, we observe ∆σm =D ∆m for all permutations σ :N →N, from whichexchangeability of the increments follows. The representation by a unique probabilitymeasure ν on EL follows from Proposition 6.2.

COMBINATORIAL LEVY PROCESSES 23

Almost sure existence of ‖X ‖ follows from Proposition 6.2 and countable additivityof probability measures. Finally, the independent increments of X implies independent,identically distributed increments for ‖X ‖, from which the Markov property is immediate.

�

7.2. Continuous-time combinatorial Levy processes. We first establish the Levy propertyfor the finite restrictions of any combinatorial Levy process.

Proof of Proposition 4.6. The finite restrictions of any combinatorial Levy process must alsohave stationary, independent increments and cadlag sample paths, by the usual charac-terization of stochastic processes through their finite restrictions and the definition of theincrement operator. Conversely, suppose X = (Xt, t ≥ 0) is a stochastic process on LNwhose finite restrictions are finite state space Levy processes. Then X has cadlag paths by thedefinition of the product discrete topology. Moreover, the increments of X are determinedby the sequence of finite state space increments, so that the stationary and independentincrements properties must also hold for X. �

Proof of Corollary 4.7. By Proposition 4.6, each of the finite state space sample paths of X isalso a Levy process. Thus, each X[n] = (Xt|[n], t ≥ 0) has stationary, independent incrementsand cadlag sample paths. For every n ∈N, L[n] is a finite state space, and the cadlag pathsassumption implies that X[n] has strictly positive hold times in all states it visits. By theStone–Weierstrass theorem for compact Hausdorff spaces,

C = {g : LN → R : there exists n ≥ 1 such that M|[n] = M′|[n] implies g(M) = g(M′)}

is dense in the space of continuous, bounded functions LN → R. The Feller propertyfollows readily. �

Let X = (Xt, t ≥ 0) be a combinatorial Levy process on LN. By Corollary 4.7, X has theFeller property and, therefore, its transition law is determined by the infinitesimal jumprates

Q(M, dM′) := limt↓0

1tP{Xt ∈ dM′ | X0 = M}.

By the stationary increments property, the jump rate from M into dM′ depends only on theincrement 4(M,M′). Thus, we can define a measure

(32) µ(d∆) :={

Q(0LN, d∆), ∆ , 0L

N,

0, ∆ = 0LN.

Proposition 7.1. The measure µ defined in (32) satisfies (18).

Proof. The lefthand side of (18) is plain by (32). The righthand side follows from Proposition4.6 as we now show. By construction, X[n] has infinitesimal jump rates

Qn(M,M′) := Q(M∗, {M′′ ∈ LN : M′′|[n] = M′}), M′ ,M ∈ L[n],

for any M∗ ∈ {M′′ ∈ LN : M′′|[n] = M}, for each n ∈ N. We interpret Qn(M,M′) as the rateat which X[n] jumps from M to M′, and so we put Qn(M,M) = 0 for all M ∈ L[n]. Since L[n]is finite and Qn(M,M′) < ∞ for all M′ ,M, we have

∞ > Qn(M,L[n]) = µ({M∗ ∈ LN : M∗|[n] , 0L[n]})

and the righthand side of (18) holds. �

24 HARRY CRANE

Proof of Theorem 4.10. Let X = (Xt, t ≥ 0) be a combinatorial Levy process with infinitesimaljump measure µ defined in (32). Let X∗µ = (X∗t , t ≥ 0) be a µ-canonical Levy process, as in

Definition 4.9. Since X∗µ is constructed from the finite state space processes X∗[n]µ , its jump

rates are determined by

µ(n)(∆) =

µ({M ∈ LN : M|[n] = ∆}), ∆ , 0L[n],

0, ∆ = 0L[n] .

We define µ∗ on events of the form {M∗ ∈ LN : M∗|[n] = M}, for M ∈ L[n] \{0L[n]}, for everyn ∈N, by

µ∗({M∗ ∈ LN : M∗|[n] = M}) = µ(n)(M).

Such events comprise a generating π-system of the Borel σ-field on LN and µ∗ is additiveby construction. Caratheodory’s extension theorem implies a unique extension of µ∗ toLN \{0LN}. Putting µ∗({0L

N}) = 0 gives a unique measure on LN. Since µ∗ coincides with µ

on the generating π-system, we must have µ∗ = µ. This completes the proof. �

We can now immediately deduce our main Levy–Ito characterization for exchangeablecombinatorial Levy processes, which we restate for the reader’s convenience.

Theorem 4.14. Let L = (i1, . . . , ik) be any signature and X = (Xt, t ≥ 0) be an exchangeablecombinatorial Levy process on LN. Then there exists a unique measure ν0 on EL satisfying

(33) ν0({0L}) = 0 and∫EL

(1 −W({0L[ik]}))ν0(dW) < ∞,

and measures µλ on LN satisfying (24), (25), and (26) such that

(34) µ = ν∗0 +∑

q=1,...,ik

∑λ`q

µ∗λ,

where λ ` q denotes that λ is a partition of q and µ∗λ is defined in (27).

Proof of Theorem 4.14. The proof follows from Theorems 4.5, 4.10, and 6.3. �

7.3. The limiting process ‖X ‖. Theorem 4.15 characterizes the existence and behavior ofthe limiting process ‖X ‖. The theorem has several parts. We first show that the projection ofan exchangeable Levy process X = (Xt, t ≥ 0) on LN exists almost surely at all time points.We then show that ‖X ‖ is itself a Feller process whose discontinuities we characterizein terms of the jumps of the covering process X. Finally, we show that ‖X ‖ has locallybounded variation almost surely.

7.3.1. Existence. By exchangeability of X, Proposition 6.2 implies that ‖Xt‖ exists for anycountable collection of times. This argument does not generalize to existence at all times t,as there are uncountably many of them. To prove existence of ‖X ‖ simultaneously at alltimes, we show that it exists at all t ∈ [0, 1] with probability 1. We deduce existence for allt ∈ [0,∞) by countable additivity and time homogeneity.

COMBINATORIAL LEVY PROCESSES 25

For every m ∈N, we define the upper and lower homomorphism densities of A ∈ L[m]in M ∈ LN by

δ+(A; M) := lim supn→∞

1n↓m

∑ϕ:[m]→[n]

1{M|ϕ[n] = A} and

δ−(A; M) := lim infn→∞

1n↓m

∑ϕ:[m]→[n]

1{M|ϕ[n] = A}.

We define the upper and lower combinatorial limits, respectively, by

‖M‖+ := (δ+(A; M), A ∈⋃

m∈N

L[m]) and

‖M‖− := (δ−(A; M), A ∈⋃

m∈N

L[m]).

Since the limits inferior and superior always exist, the upper and lower limits of M arewell defined. The limit ‖M‖ exists if and only if ‖M‖+ = ‖M‖−, so we must show that‖X‖+ = (‖Xt‖

+, 0 ≤ t ≤ 1) and ‖X‖− = (‖Xt‖−, 0 ≤ t ≤ 1) coincide with probability 1.

By the canonical construction of X from a time homogeneous Poisson point process ∆∗

with intensity dt ⊗ µ, there is probability 0 of a discontinuity at any given time t ∈ [0, 1]. Bythe cadlag paths assumption, each finite component X[n] of X experiences at most finitelymany discontinuities in [0, 1]. Thus, for every m ∈N and every ε > 0, there is a partition of[0, 1] into finitely many non-overlapping subintervals J1, . . . , JK such that

P{∆∗ has an atom (t,∆t) in Jl for which ∆t|[m] , 0L[m]} < ε for every l = 1, . . . ,K.

Since the action of relabeling is ergodic for exchangeable processes, the law of large numbersimplies

limn→∞

1n↓m

∑ϕ:[m]→[n]

1{(Xt|ϕ[n], 0 ≤ t ≤ 1) is discontinuous on Jl} < ε,

for every l = 1, . . . ,K. Thus, for each l = 1, . . . ,K, the upper and lower homomorphismdensities for any A ∈ L[m] cannot vary by more than ε. Furthermore, the upper and lowerdensities are equal on the fixed set of endpoints of J1, . . . , JK, implying

supt∈[0,1] |δ+(A; Xt) − δ−(A; Xt)| ≤ 2ε a.s.,

for every A ∈ L[m], for every m ∈ N. Since ε > 0 is arbitrary, it follows that δ+(A; Xt) =δ−(A; Xt) simultaneously for all t ∈ [0, 1] with probability 1. Countable additivity impliesthat ‖Xt‖

+ = ‖Xt‖− simultaneously for all t ∈ [0, 1] with probability 1.

7.3.2. The Feller property. Every combinatorial limit ‖M‖determines a combinatorial semigroupP = (P(A,A′),A,A′ ∈

⋃n∈N L[n]), defined by

(35) P(A,A′) =

{‖M‖(A4A′), A,A′ ∈ L[n], for some n ∈N,

0, otherwise.

We write SL to denote the space of combinatorial semigroups corresponding to LN. Thefollowing is an immediate consequence of Proposition 6.2.

26 HARRY CRANE

Proposition 7.2. Given an exchangeable probability measureµ onLN, the semigroup P constructedin (35) is a collection of exchangeable transition probability measures onLN. Moreover, each P ∈ SLacts on SL by P′ 7→ P ◦ P′, where

(P ◦ P′)(A,A′) :=∑

A′′∈L[n]

P′(A,A′′)P(A′′,A′), A,A′ ∈ L[n],

for every n ∈N.

Since SL consists of exchangeable transition probability measures and EL consists ofexchangeable probability measures, both on LN, any P ∈ SL acts on µ ∈ EL by

(36) (P ◦ µ)(A) = (Pµ)(A) :=∑

A′∈L[n]

µ(A′)P(A′,A), A ∈ L[n], for every n ∈N .

Definition 7.3 (Semigroup process). Given a combinatorial Levy process X = (Xt, t ≥ 0) onLN,we define its associated semigroup process P = (Pt, t ≥ 0) on SL by

Pt(A,A′) =

{‖Xt 4X0‖(A4A′), A,A′ ∈ L[n], for some n ∈N,

0, otherwise, t ≥ 0,

if it exists.

To prove the Feller property for ‖X ‖, we define the metric

d(W,W′) :=∑n∈N

2−n∑

A∈L[n]

|W(A) −W′(A)|, W,W′ ∈ EL,

under which EL is compact. The semigroup operation in (36) is Lipschitz continuous inthis metric.

Existence of the semigroup process P = (Pt, t ≥ 0) follows from existence of ‖X ‖ forevery exchangeable combinatorial Levy process X, by Theorem 4.15. We say that P hasstationary, independent increments if Pt+s =D P′t ◦ Ps, for all s, t ≥ 0, where P′ = (P′t, t ≥ 0) isan independent, identically distributed copy of P. These aspects of P follow by precedingarguments, as the increments of P are directly defined in terms of the increments of thecovering process X, which are stationary and independent. Cadlag sample paths followsfrom cadlag sample paths of ‖X ‖.

By the Poisson point process construction of X, we can couple X, ‖X ‖, and P so that‖Xt‖ = Pt ◦ ‖X0‖ for all t ≥ 0. The Feller property of ‖X ‖ implies that Pt ↓ P0 as t ↓ 0and, thus, Pt ◦ ‖X0‖ → ‖X0‖ by Lipschitz continuity of the semigroup action. Thus, forany bounded, continuous function g : EL → R, the bounded convergence theorem andcontinuity of g imply

limt↓0

E(g(‖Xt‖) | ‖X0‖) = limt↓0Eg(Pt ◦ ‖X0‖) | ‖X0‖)

= E(limt↓0

g(Pt ◦ ‖X0‖) | ‖X0‖)

= E(g(limt↓0

Pt ◦ ‖X0‖) | ‖X0‖)

= E(g(‖X0‖) | ‖X0‖)= g(‖X0‖).

The second part of the Feller property, that x 7→ E(g(‖Xt‖) | ‖X0‖ = x) is continuous forevery t > 0, follows by continuity of g and Lipschitz continuity of the semigroup action.

COMBINATORIAL LEVY PROCESSES 27

Specifically, fix W ∈ EL and let (Wn,n ≥ 1) be a sequence in EL such that Wn →W. For anybounded, continuous g : EL → R, we have

limn→∞

E(g(‖Xt‖) | ‖X0‖ = Wn) = limn→∞

E(g(Pt ◦ ‖X0‖) | ‖X0‖ = Wn)

= limn→∞

E(g(Pt ◦Wn))

= E(g(Pt ◦W)),

because the composition of bounded, continuous functions g and Pt is bounded andcontinuous.

7.3.3. Characterization of discontinuities. By time homogeneity and cadlag sample paths,there is 0 probability that X[n] experiences a discontinuity at any given t ∈ [0, 1]. Moreover,every X[n] can experience only finitely many discontinuities in [0, 1] and X experiencesat most countably many discontinuities in [0, 1]. In Theorem 4.14, we characterized thediscontinuities of X by decomposing its jump measure µ into mutually singular componentsindexed by the subsignatures of L. To characterize the discontinuities of ‖X ‖, we proceedsimilarly. In particular, we observe that the discontinuity times of ‖X ‖ must be a subsetof the discontinuity times of X. By Theorem 4.14, cf. Theorem 6.3, we call t a type-qdiscontinuity if |∩ {s′ : ‖M‖s′ , 0}| = q. In particular, a type-0 discontinuity is a time t ∈ [0, 1]for which ‖Xt− 4Xt‖ , 0L . The theorem asserts that ‖X ‖ is continuous except possibly atthe times of type-0 discontinuities in X. We treat the cases q = 0 and q = 1, . . . , ik separately.

Case I: q = 0. In this case, Definition 3.6 implies

limn→∞

1n↓m

∑ϕ:[m]→[n]

1{Xt−|ϕ[n] , Xt|

ϕ[n]} > 0, for every m ∈N,

which permits the possibility that ‖X ‖ experiences a discontinuity at time t.

Case II: q = 1, . . . , ik. To be concrete, we assume q is a fixed integer between 1 and ik, whichimplies there is some unique s = {s1, . . . , sq} ⊆N for which X can experience a discontinuityat time t in potentially all entries indexed by a with s ⊆ a. There are at most nm−q suchmultisets of [n] for each m ≥ q. Thus,

0 ≤ limn→∞

1n↓m

∑ϕ:[m]→[n]

1{Xt−|ϕ[n] , Xt|

ϕ[n]} ≤ lim

n→∞

nm−q

n↓m= 0,

because q ≥ 1. Thus, ‖Xt−‖ = ‖Xt‖ and ‖X ‖ is continuous at type-q discontinuity times, forall q ≥ 1.

8. Concluding remarks

Stochastic process models for dynamic combinatorial structures have a place in DNAsequencing, dynamic network modeling, combinatorial search algorithms, phylogenetics,and much more. They also have potential for modeling certain composite structures, asdiscussed in Section 5.3. With an array of applications in mind, we have developed thetheory of combinatorial Levy processes in its most general setting and proven severalfundamental theorems about their behavior. We foresee potential of combinatorial Levyprocesses for modeling many other dynamic structures.

In the specific case of network modeling, recent efforts, for example, [6, 11, 35], oftenfocus on a specific phenomenon or application to which a particular model can be tailored.

28 HARRY CRANE

In such situations, domain specific knowledge leads to further insights about structureand function of the underlying network. None of the above cited models incorporatestemporal dynamics in the way we have considered here. Our discussion lays out a generalprescription and our main theorems guarantee nice behavior under basic sample pathproperties.

Acknowledgment

The author’s work is partially supported by NSF grants CNS-1523785 and CAREERDMS-1554092.

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Rutgers University, Department of Statistics, 110 Frelinghuysen Road, Piscataway, NJ 08854.