The deFinetti representation of generalised Marshall–Olkin sequences

We show that each in nite exchangeable sequence τ1, τ2, . . . of randomvariables of the generalised Marshall–Olkin kind can be uniquely linked to an additive subordinator via its deFinetti representation. This is useful for simulation, model estimation, and model building.

In [21], the authors proposed the exogenous shock model as a natural stochastic model for the Marshall-Olkin distribution. This model is based on independent, exponentially distributed random times corresponding to the failure of multiple components of a system at once. In particular, for λ I ≥ , ∅ ≠ I ⊆ { , . . . , d}, ful lling the condition in Eq. (2), let E I ∼ Exp(λ I ) be independent exponentially distributed random variables with rates λ I , ∅ ≠ I ⊆ { , . . . , d}, respectively, where we use the convention that an exponentially distributed random variable with rate zero is almost surely in nite. De ne τ = (τ , . . . , τ d ) by Then τ has a Marshall-Olkin distribution with parameters λ I , ∅ ≠ I ⊆ { , . . . , d}.
We are interested in exchangeable random vectors and sequences of (generalised) Marshall-Olkin kind. These subclasses have been intensively studied in the last decade for the classical Marshall-Olkin distribution.
Furthermore, he has shown that the survival function in Eq. (1) of an exchangeable Marshall-Olkin distribution can be reparametrised as follows: where t [ ] ≥ · · · ≥ t [d] is t in descending order. The sequence a , a , . . . a d− is de ned by where λ i = λ I for i = |I|. Finally, he provides a characterisation theorem that states that a functionF of the form of Eq. (4) is a survival function if, and only if, the sequence a , a , . . . , In this case, the author shows that • The extendible subclass is studied in [11,Chp. 4] where E i are iid unit exponential random variables independent of a Lévy subordinator {Λ t } t≥ on [ , ∞]. This model is also called the Lévy frailty model. A natural generalisation of the classical Marshall-Olkin distribution is achieved if we allow non-constant hazard rates in the exogenous shock model construction in Eq. (3), see [10]. This means that we replace λ I · max i∈I t i with a cumulative hazard rate function H I (max i∈I t i ) and the exponential shocks . . , d}, respectively. A cumulative hazard rate function is a nonnegative, non-decreasing, and continuous function on the non-negative half-line that starts in zero. Previous works exist on special cases of this generalisation, e.g. [9], which discusses the bivariate case, and [22], which assumes that H I (t) ≡ λ I H(t).
• The exchangeable generalised Marshall-Olkin distribution and the exchangeable exogenous shock model are studied in [25]. Similar to the classical Marshall-Olkin case, the author has proven that exchangeability corresponds to the property Furthermore, he has shown that a reparametrisation is possible, similar to the classical Marshall-Olkin case, by replacing . . , d}, in Eq. (4). He also provides an analytical characterisation, which is discussed in Section 3. • The extendible subclass of generalised Marshall-Olkin distributions is studied in [14] and [25,Sec. 3].
In [14,Prop. 3.1], it is shown that if the subordinator Λ in Eq. (5) is assumed to be an additive subordinator in [ , ∞], then each nite margin of {τ i } i∈N has an extendible generalised Marshall-Olkin distribution.
We call this stochastic model the additive-frailty model.

Contribution:
This article provides the following novel result, which was posed as an open problem for further research in [25, p. 147 sq.]: every exchangeable sequence τ , τ , . . . with nite margins of generalised Marshall-Olkin type has an implicit representation as an additive frailty model. In particular, an additive subordinator Λ and an iid sequence of unit exponential random variables E , E , . . ., independent of Λ, exist such that Eq. (5) is ful lled almost surely. Recall that the converse of this statement was proven in [14,Prop. 3.1]. Consequently, we complete this result and establish a novel one-to-one connection between sequences of generalised Marshall-Olkin type and additive subordinators. The article is structured as follows: we introduce the mathematical background and notation in Section 2, we summarise existing results on exchangeable generalised Marshall-Olkin distributions in Section 3, and we present the main result in Section 4. In Section 5, we conclude the article. The main proof requires some technical results involving exchangeable sequences and Bernstein functions. For the interested reader, we summarise the required background in Appendices A and B.

Mathematical background and notation
In this section, we give a short overview of the required mathematical background and the used notation.
We assume basic knowledge of the theory on multivariate distribution functions and probability theory. Furthermore, we assume that the reader is familiar with the Lévy-Khintchine characterisation of additive subordinators. Additive processes are real-valued, stochastic processes, which are de ned on the non-negative half-line, start at zero, have independent increments, and have càdlàg path. An additive subordinator is a nondecreasing additive process which tends almost surely to in nity. Excellent books on additive processes and Lévy processes in particular are [2,24]. We deviate slightly from the standard theory by allowing the additive subordinator to jump to an absorbing point associated with ∞ at a random time, which is independent from the subordinator. The corresponding (cumulative) hazard rate is called (cumulative) killing hazard rate and is equal to the zero function if almost surely no killing occurs. The Lévy-Khintchine characterisation states that each additive subordinator is uniquely determined in law by its family of Laplace exponents. These Laplace exponents are from the family of Bernstein functions, hereafter denoted by BF. A function ψ : ( , ∞) → ( , ∞) is a Bernstein function if it is in nitely often di erentiable and has the following property One can show, see e.g. [ Here, ∆ is the forward iterated di erence operator. A Bernstein function ψ is assumed to be extended to the domain [ , ∞) by the convention ψ( ) = . Excellent books on Bernstein functions are [3,26]. We denote random variables with capital or Greek letters, e.g. X or τ , and (random) vectors with bold letters, e.g. X, τ , or t. We write X ∼ F if X has the distribution function F. We assume that operators are applied component-wise to vectors. That means τ > t is equivalent to We denote the class of continuous, real functions by C ( ) , we write ∆f ≥ if the function f is nondecreasing everywhere, and we use the notation f (x−) := lim y x f (y) as well as f (x+) := lim y x f (y). Finally, for a real number x, we denote the smallest integer i with i ≥ x by x .

Exchangeable generalised Marshall-Olkin distributions
In this section, we give a short introduction into exchangeable generalised Marshall-Olkin distributions. For a more detailed treatment of the exchangeable subclass, see [14,25].
We generalise the classical Marshall-Olkin distribution by allowing arbitrary continuous, cumulative hazard rate functions in the exogenous shock model. This is equivalent to having continuous, non-negative, and unbounded shock-times. For this, we de ne the class of continuous, cumulative hazard rate functions H and its unbounded subclass H by We say that a random vector τ ∈ [ , ∞) d has a generalised Marshall-Olkin distribution if functions H I ∈ H, and the hazard rate functions ful l the condition This condition, which generalises the condition in Eq. (2), is equivalent to the margins being almost surely nite, since I i H I are the marginal cumulative hazard rates. With a simple calculation, we can establish a generalised version of the exogenous shock model in Eq. (3) for generalised Marshall-Olkin distributions by replacing λ I · max i∈I t i with H I (max i∈I t i ) and Below, we present a characterisation of exchangeable generalised Marshall-Olkin distributions. We know from [14, Prop. 2.1] that, similar to the classical Marshall-Olkin case, exchangeability is equivalent to the property H I = H J for all ∅ ≠ I, J ⊆ { , . . . , d} with |I| = |J|. Furthermore, the following characterisation result has been proven in [14]: is t in descending order. Then the following statements are equivalent: 1.F is the survival function of a random vector on [ , ∞) d .

It holds that H
Finally, we can construct a random vector τ with survival functionF via an exogenous shock model with Proof of Lemma 1. This is a direct corollary of [14, Thm. 1.1]. However, since we changed the notation, we will give a short explanation: if we take the standardisation of the margins into account, the aforementioned result a rms that the rst statement of this lemma is equivalent to

The deFinetti representation of GMO sequences
In this section, we characterise the deFinetti representation of exchangeable generalised Marshall-Olkin sequences. We begin with an overview of general deFinetti representations. We know from deFinetti's theorem, see [1,Thm. 3.1], that an almost surely unique random distribution function F exists for each exchangeable sequence τ , τ , . . . such that almost surely For a non-decreasing function h, we de ne its [7] for a detailed discussion of generalised inverses. If the random distribution function F has almost surely no jumps, we have that almost surely for an iid uniform sequence U , U , . . ., independent of F, which is de ned by we can rewrite Eq. (7) as for a (càdlàg) subordinator Λ and a sequence E , E , . . . of iid unit exponential random variables, independent of Λ. For this, we de ne Λ = − log ( − F) and E i = − log ( − U i ), i ∈ { , . . . , d}. Note that we de ne a subordinator as a [ , ∞]-valued, non-decreasing, càdlàg process on [ , ∞) that starts at zero and tends to in nity for t → ∞. If the random distribution function F may possibly have jumps, then Eqs. (7) and (8) still hold if there is an additional iid uniform sequence W , W , . . ., which is independent of τ , τ , . . ., de ned on the probability space. The sequence W , W , . . . is required to modify F(τ i ) to a uniform random variable by a random interpolation at its (random) atoms, see [23].
With random hazard rate Before moving on to the main result of this article, we want to outline three applications of the deFinetti representation: 1. We can use the deFinetti representation to sample from certain distributions e ciently in highdimensions as illustrated in Fig. 1. See, e.g., [11,16,19] for applications of this technique. 2. We can build low-parametric, dimensionless families of multivariate distributions from parametrised subordinators, see, e.g., [4,15,17] for examples. We call these families dimensionless, since a random vector from such a model can be de ned as the margin of an in nite sequence. Consequently, these families are not inherently linked to a speci c dimension.
3. We can use the deFinetti representation for exchangeable sequences to build hierarchical models for non-exchangeable sequences. We refer the interested reader to [12,18,20]. Below, we state the main result of this article and investigate the subordinator, which is implied by the de-Finetti representation of generalised Marshall-Olkin sequences. We already know from [11] that Marshall-Olkin sequences are uniquely linked to Lévy subordinators via Eq. (8). That remains true if we generalise the Marshall-Olkin de nition as in Section 3 and generalise the Lévy subordinator to an additive subordinator.
Furthermore, assume that an iid uniform sequence W , W , . . . which is independent of τ , τ , . . . is de ned on the probability space. Then, an additive subordinator Λ and iid unit exponentially distributed random variables E , E , . . ., independent of Λ, exist such that almost surely

Proof. Firstly, note that the backward direction is a corollary of [14, Prop. 3.1] by considering marginal transformations.
For the forward direction, which is the main contribution of this article, we use deFinetti's theorem, see [1, Thm. 3.1], to obtain the existence of a random distribution function F such that the sequence is conditionally iid given F and Eq. (6) holds. We de ne We use [23,Sec. 2] to obtain that, conditioned on F, U , U , . . . are uniform and ful l almost surely Eq. (6). In particular, we have that almost surely In summary, the sequence U , U , . . . is iid uniform and independent of F. We use the transformations Λ = − log ( − F) and E i = − log ( − U i ), i ∈ N and obtain a subordinator Λ and an iid unit exponential sequence E , E , . . ., independent of Λ, such that Eq. (10) holds almost surely. Now, we have to prove that Λ is an additive subordinator. By a simple uniqueness-in-distribution argument and [25, p. 41], we determine that this is equivalent to the existence of a family of Bernstein functions {ψ t } t≥ ⊆ BF, ful lling the conditions such that Eq. (11) holds. Below, we show that a family of Bernstein functions with these properties exists.
With Lemma 1, we have Fix s > t ≥ . Then, we have for arbitrary d ∈ N

This implies that the sequence A (s) − A (t), A (s) −
This implies Thus, if we set ψu = ψ u, for u ∈ {t, s}, we obtain ψ ≡ and We use the fact that Bernstein functions are determined by their values on N , see [3,Prop. 6.12] and [26,Thm. 3.2], and we get ψs − ψ t = ψ s,t ∈ BF. Finally, we use that a Bernstein function is non-negative and monotone increasing to obtain the following formula for s > t ≥ and x ≥ that Hence, the continuity of A , A , . . . implies lim t k →t ψ t k (x) = ψ t (x) for all t, x ≥ .

Recovery of the subordinator
Theorem 1 motivates the following questions: rstly, what are non-trivial examples of how the forward direction of this theorem can be used and secondly, how can we use the theorem to learn more about the implied subordinator. A non-trivial example is an exchangeable, but not comonotone or independent, generalised Marshall-Olkin sequence, which is not directly generated by a deFinetti model. Given such a sequence, the theorem only guarantees the existence of a deFinetti representation, but does not explicitly state the law of the subordinator or how it can be explicitly recovered. In the following, we use an example adapted from [13,Expl. 6.3] to demonstrate how the subordinator can be identi ed and recovered. We consider an exogenous shock model in which each component can fail due to independent individual shocks or a common global shock. For this, let H, H G ∈ H with H + H G ∈ H be cumulative hazard rate functions and de ne A = H and A = H + H G . Furthermore, let Z G ∼ − exp {−H G } and let Z , Z , . . . be an iid sequence with distribution function − exp {−H} that is independent of Z G . We de ne the random sequence τ , τ , . . . by

Recovery of the subordinator law
In the rst step, we use the generalised version of the exogenous shock model representation from Eq. (3) and the novel result from Theorem 1 to determine that the subordinator, implied by the deFinetti representation, is an additive subordinator with cumulative killing hazard rate H G and deterministic part H. Since the sequence τ , τ , . . . is exchangeable and of generalised Marshall-Olkin kind, we know that the random vector τ = (τ , . . . , τ d ) has an exchangeable generalised Marshall-Olkin distribution for each d ∈ N. We use Lemma 1 and determine that the corresponding survival function is Then, we conclude with Theorem 1 that an additive subordinator Λ with the characterising family of Bernstein functions {ψ t } t≥ exists such that This implies for t ≥ that As Bernstein functions are uniquely de ned by their values on N , we verify that This family of Bernstein functions can be identi ed with an additive subordinator with (inhomogeneous) cumulative killing hazard rate H G (t) and deterministic part H(t). In particular, a random variable Z ∼ − exp {−H G } exists such that Note that so far, we only know that some random variable Z exists such that this equation holds. A natural conjecture is that Z = Z G , which is proven in the following.

Explicit recovery of the subordinator
In the second step, to derive the subordinator explicitly, we use that for the tail-σ-algebra T of the sequence τ , τ , . . . and a P-nullset N, see [1, Lem. 2.15 and 2.19]. Furthermore, we use that Z j > t for in nitely many j and therefore Consequently, Z G is measurable with respect to T. Moreover, we have for ω ∈ Ω\N where we use the convention that · ∞ = . Finally, given an iid uniform sequence W , W , . . ., independent of τ , τ , . . ., we can construct the sequence E , E , . . . by Now, Theorem 1 implies that the sequence E , E , . . . is iid unit exponential, independent of F, and we conclude that almost surely

Conclusion
We have shown that exchangeable sequences τ , τ , . . . of a generalised Marshall-Olkin kind are uniquely linked to additive subordinators via a deFinetti representation. In particular, in a suitably extended probability space, we have almost surely that where Λ is an additive subordinator and the sequence E , E , . . . is iid unit exponential and independent of Λ.
Acknowledgements: Many thanks to Lexuri Fernández, Florian Brück, Matthias Scherer, and the two anonymous reviewers for their feedback on this article.

A Exchangeable sequences and DeFinetti's theorem
In this section, we summarise the background on exchangeable sequences and deFinetti representations. An extensive reference on the deFinetti representation of exchangeable sequences and exchangeability in general, which contains all results that are presented in this section, is [1]. We call a sequence τ , τ , . . . exchangeable if In the following, we show how the directing measure α can be calculated from the sequence τ , τ , . . .. For this, assume that the sequence τ , τ , . . . is de ned in the probability space (Ω, F, P) and let T be its tail-σalgebra. On existence, α is a.s. unique, T measurable, and a regular conditional distribution for τ given T, see [1,Lem. 2.15 and 2.19]. Thus, we have where N is a P-nullset. In the following, we assume w.l.o.g. that α(ω, A) = for all ω ∈ N, A ∈ B. Finally, since α(ω, ·) is a (random) probability measure on R, we may identify α(ω, ·) with a random distribution function If another sequence of iid uniform random variables W , W , . . ., which is independent of τ , τ , . . . is de ned on the probability space, we can re ne deFinetti's theorem: Finally, with the de nition of the regular conditional distribution, we establish that U , U , . . . is an iid uniform sequence that is independent of T, hence also independent of F.
We conclude this section with an example that explains the need for additional randomness, in form of an iid uniform sequence W , W , . . ., in the two preceding theorems. This example also highlights that not every conditionally independent sequence has a representation as in Eq. (13) when only the original probability space is considered. For this, let (Ω, F, P) be the Lebesgue probability space on the interval [ , ] and de ne Clearly, the sequence U , U , . . . is exchangeable and U is measurable with respect to the sequences tailσ-algebra T. Hence, we can calculate the random distribution function F, corresponding to the sequences directing measure α, for all ω excluding a Lebesgue-nullset and u ∈ [ , ] by Since σ(F) = σ(U ) = F, there is no additional iid sequence independent of F de ned on this probability space. If we now consider the enclosing probability product space, on which U as well as an iid uniform sequence W , W , . . ., independent of U , are de ned, we have

B Bernstein functions and completely monotone sequences
The proof of the main theorem relies heavily on the connection between additive and Lévy subordinators, so-called Bernstein functions, and completely monotone sequences. As the topic cannot be treated in detail without using deep results of functional analysis and measure theory, we will limit ourselves to presenting the main results. Extensive references on this topic are [3,26]. Another excellent reference is [11,Chp. 3 and 4].
We denote the set of all Bernstein functions by BF and use the convention that a Bernstein function may be extended to [ , ∞) by setting ψ( ) := . It is well-known, see, e.g. where we call a measure ν on ( , ∞) a Lévy measure if ( ,∞) ( ∧ x) ν(dx) < ∞. In that case (a, b, ν) is uniquely determined by ψ and is called the Lévy triplet.
Bernstein functions ψ with ψ( ) = can be uniquely linked to so-called completely monotone sequences. For a (countably in nite) sequence a , a , . . ., let ∆ be the discrete di erence operator de ned by ∆a i := a i+ − a i and de ne recursively ∆ n a i := ∆(∆ n− a i ). We call the sequence a , a , . . . completely monotone if We call a nite sequence a , . . . , a d− d-monotone if (− ) i ∆ i a k ≥ , ∀i, k ∈ N : i + k < d.