A note on the Galambos copula and its associated Bernstein function

Abstract There is an infinite exchangeable sequence of random variables {Xk}k∈ℕ such that each finitedimensional distribution follows a min-stable multivariate exponential law with Galambos survival copula, named after [7]. A recent result of [15] implies the existence of a unique Bernstein function Ψ associated with {Xk}k∈ℕ via the relation Ψ(d) = exponential rate of the minimum of d members of {Xk}k∈ℕ. The present note provides the Lévy–Khinchin representation for this Bernstein function and explores some of its properties.


Introduction
A d-dimensional random vector (X , . . . , X d ) follows a min-stable multivariate exponential law (MSMVE) if min{c X i , . . . , c k X i k } has a (univariate) exponential law for all ≤ i < . . . < i k ≤ d and constants c , . . . , c k > . This is the case if and only if its components have (univariate) exponential laws and its survival copula is of extreme-value kind, see [13,Theorem 6.2,p. 174]. A sequence of random variables {X k } k∈N on a probability space (Ω, F, P) is said to be MSMVE if min{c X i , . . . , c k X i k } has a (univariate) exponential distribution for arbitrary c , . . . , c k > and i < i < . . . < i k , k ∈ N. Provided such a sequence is in addition exchangeable, i.e. the law of {X σ(k) } k∈N is invariant under bijections σ : N → N, the article [15] provides a canonical stochastic construction by means of non-decreasing stochastic processes which are strong in nitely divisible with respect to time (strong IDT), a notion that was introduced and investigated in [6,9,16]. A stochastic process {H t } t≥ is strong IDT if for each n ∈ N one has t } t≥ are independent copies of {H t } t≥ , i ∈ N. Indeed, there is a oneto-one relationship between exchangeable MSMVE sequences and right-continuous, non-decreasing strong IDT processes starting from H = and satisfying lim t→∞ H t = ∞, which is induced by the stochastic model where {ϵ k } k∈N is an iid sequence of unit mean exponential random variables, independent of {H t } t≥ . One well-studied subfamily of non-decreasing strong IDT processes is the family of (killed) Lévy subordinators. However, there exist also proper increasing strong IDT processes that are not Lévy subordinators, and their rigorous study is an interesting topic for further research. In particular, the present article studies an exchangeable MSMVE sequence which is associated with a non-Lévy strong IDT process, whose probability law, however, is unknown.
The present note contributes the following two points to the study of the law of this unknown strong IDT process: -Section 2 explains that for each θ > there is an exchangeable MSMVE sequence such that the exponential rate of the minimum min{X , . . . , X d } is given by These rates stem from an MSMVE distribution rst introduced in [7]. -Given the rst bullet point, it follows from [15] that there exists a unique Bernstein function Ψ θ such that Ψ θ (d) = λ θ (d) for all d ∈ N. A Bernstein function is de ned on [ , ∞), non-negative, starting at zero, and is smooth on ( , ∞) with completely monotone rst derivative. For background on Bernstein functions we recommend [19]. Section 3 provides the Lévy-Khinchin representation of Ψ θ and studies its properties.

The stochastic construction of the Galambos MSMVE sequence
For the sake of simpli ed notation, we introduce the following de nition. There is a one-to-one relationship between cumulative hazard processes and in nite exchangeable sequences of random variables with support in [ , ∞), induced by the canonical construction (1), which is an immediate consequence of De Finetti's Theorem. The original references are [4,5], popular generalizations to more general state spaces are achieved in, e.g., [12,17].
Applying construction (1) with the processes H = H (n) for each n, we obtain sequences of random variables -For each k, n ∈ N the survival function of X (n) k is given by In particular, as n → ∞, the law of X (n) k converges weakly to a unit exponential law. -If Cφ denotes the Archimedean copula (see, e.g., [3] for background on the latter) generated by φ, i.e.
By [3, Theorem 3.1], the sequence converges to The function θ in (5) is a so-called stable tail dependence function associated with an MSMVE in the sense that exp(− θ (t , . . . , t d )) is the survival function of an MSMVE distribution. Alternatively, ) is an extreme-value copula. More precisely, it corresponds to the socalled Galambos copula, named after [7]. Since we have for arbitrary sequences {an} n∈N that lim n→∞ + an n n = e a ⇔ lim n→∞ an = a, we conclude i.e. the distribution of (X (n) , . . . , X (n) d ) converges to the Galambos MSMVE as n → ∞, for arbitrary d ∈ N. Concluding, we have a sequence of in nite sequences of random variables, such that the nitedimensional distributions of them converge weakly to Galambos MSMVEs.
-As a special case of the previous bullet point, we also get for m , . . . , m d ∈ N and t , . . . , t d ≥ arbitrary that .
Since the (multivariate) Laplace transforms of nite-(d-)dimensional distributions from {H (n) t } are completely determined by their values on N d by the Theorem of Stone-Weierstrass, we can conclude that the following statement holds true.

Lemma 2.2 (Existence of the Galambos strong IDT process). There is a strong IDT cumulative hazard process H such that
In particular, the exchangeable sequence {X k } k∈N de ned via (1) from the process H has Galambos MSMVEs as nite-dimensional distributions.
Proof. The computations above imply that the cumulative hazard processes H (n) converge weakly to some cumulative hazard process H, whose nite-dimensional distributions are given as claimed in the statement. Now H (n) is of the structural form  (1). Notice in particular that H cannot be a jump process, because its jumps would induce positive probabilities for events such as {X = X }, which is not the case for the Galambos MSMVE. As a rst step into studying the stochastic behavior of H the next section studies the in nitely divisible law of H t for xed t > . Figure 1 shows a bivariate scatter plot which should approximately resemble a scatter plot from the Galambos copula (u , u ) → exp(− θ (− log(u ), − log(u ))) for θ = . It is generated by simulating iid samples of the random vector (X (n) , X (n) ) from the canonical construction (1) with the approximating cumulative hazard process (3), for n = , and then transforming to uniform marginals by applying the survival function (4) to both components, i.e. visualized are samples of (U (n) , U (n) ) := max ( − X (n) /n) n , , max ( − X (n) /n) n , .
The involved Laplace transform has been speci ed as φ(x) = ( + x) − /θ with θ = , resulting in a unit exponential distribution for the involved random variables M , M , . . .. Furthermore, based on N := iid samples of (U (n) , U (n) ) for varying n, denoted (U (n) (k), U (n) (k)) for k = , . . . , N, we computed the empirical Spearman's Rhoρ and compared it with the theoretical Spearman's Rho of the limiting Galambos copula, which is given by see, e.g., [8]. It is observed that the empirical value for Spearman's Rho approximates the theoretical value with increasing n, as implied by the theory.

The Galambos Bernstein function
In the previous paragraph we have constructed an exchangeable MSMVE sequence {X k } k∈N and a nondecreasing, strong IDT process H such that It is known from [15] that there exists a Bernstein function Ψ θ such that the law of H t is in nitely divisible with Laplace transform exp(−t Ψ θ ), each t ≥ . However, according to [15] this Bernstein function is not known in  closed form except for the case θ = . The only thing that is known is that the exponential rate of the minimum min{X , . . . , X d } is given by The following proposition derives the Lévy-Khinchin representation of the Bernstein function Ψ θ for arbitrary θ > . To this end, recall that any Bernstein function Ψ has a Lévy-Khinchin representation with a constant b ≥ and a measure ν on ( , ∞] satisfying ( ,∞] min{t, } ν(dt) < ∞, called the Lévy measure associated with Ψ.

Proposition 3.1 (The Galambos Bernstein function). The Bernstein function Ψ θ associated with the Galambos MSMVE is given by the Lévy-Khinchin representation
Proof. Fix θ and consider the sequence a k := Ψ θ (k + ) − Ψ θ (k), k ∈ N . From (6) we conclude that Let τ be a Γ-distributed random variable with density fτ(x) = x θ− e −x /Γ(θ), x > . From the knowledge about the Laplace transform of the Γ-distribution it is observed that {(k + ) −θ } k∈N is the moment sequence of exp(−τ), i.e. (k + ) −θ = ∞ exp(−k x) fτ(x) dx. Consequently, implying that {a k } k∈N equals the moment sequence of the random variable X := − exp(−τ). The density f X of X is given by An application of [14,Lemma 5.3] implies that the Lévy measure of Ψ θ is given by which implies the claim.
Interestingly, (the in nitely divisible law associated with) the Bernstein function of Proposition 3.1 seems to be completely unstudied in the academic literature. The following corollary collects some properties of the Bernstein function Ψ θ and, hence, about the stochastic nature of H. . So for θ = we already observe that Ψ is complete. For θ ∈ { , , , . . .}, we show that the function t → (− log( − exp(−t))) θ− is completely monotone as well, which then implies the claim, as products of completely monotone functions are completely monotone again. If φ is completely monotone and β ∈ N, then φ β is also completely monotone since the set of completely monotone functions is closed under multiplication, which was used before. Moreover, the function t → − log( − exp(−t)) is c.m., since its rst derivative equals t → − exp(−t)/( − exp(−t)), and the latter function is the negative of a completely monotone function (as was just explained). This ultimately proves that t → (− log( − exp(−t))) θ− is completely monotone for all θ ∈ { , , , . . .}. (c) Recall that an in nitely divisible law is self-decomposable if and only if it has a Lévy measure of the form k(t) dt with t → t k(t) non-increasing, see [18,Chapter 3,. In the present situation we have k(t) = k θ (t) = exp(−t)/( − exp(−t)) (− log( − exp(−t))) θ− /Γ(θ). Furthermore, we compute which is strictly smaller than zero if and only if Now h(t) is a continuous function on ( , ∞) with lim t↓ h(t) = , which can be observed by an application of L'Hospital's rule, and h(t) < , which follows from the fact that e t (t − ) + > and log( − e −t ) < . Consequently, for θ ≥ the function t → t k θ (t) has strictly negative derivative for all t > and the law is selfdecomposable, as claimed. For θ < , however, we nd a non-empty interval on which h(t) > θ. Consequently, the function t → t k θ (t) is increasing on that interval and the law not self-decomposable.

Conclusion
It was shown that there exists a strong IDT process H based on which random vectors with Galambos survival copula can be constructed via (1). The present note has further collected some stochastic properties of H and its associated Bernstein function.
Since the polynomials are dense in the set of continuous (and bounded) functions on the compact interval [ , ], it follows that µn tends weakly to µ . This in turn implies the claim, since for arbitrary x ≥ we observe Ψn(x) = − log [ , ] y x µn(dy) −→ − log [ , ] y x µ (dy) = Ψ (x), n → ∞.