The generalized Pareto process; with a view towards application and simulation

In extreme value statistics, the peaks-over-threshold method is widely used. The method is based on the generalized Pareto distribution characterizing probabilities of exceedances over high thresholds in $\mathbb {R}^d$. We present a generalization of this concept in the space of continuous functions. We call this the generalized Pareto process. Differently from earlier papers, our definition is not based on a distribution function but on functional properties, and does not need a reference to a related max-stable process. As an application, we use the theory to simulate wind fields connected to disastrous storms on the basis of observed extreme but not disastrous storms. We also establish the peaks-over-threshold approach in function space.


Introduction
We say that a stochastic process X in C(S) (the space of continuous real functions on S, with S a compact subset of R d ) is in the domain of attraction of a max-stable process if, there are continuous functions a n (s) positive and b n (s) on S such that the processes max 1≤i≤n X i (s) − b n (s) a n (s) s∈S , with X, X 1 , . . . , X n independent and identically distributed, converge in distribution to a max-stable process Y in C(S). Necessary and sufficient conditions for this to happen are: uniform convergence of the marginal distributions and a convergence of measures (in fact a form of regular variation): converges as t → ∞ and so does The limit constitutes a probability distribution on C + (S). This reasoning is quite similar to how one gets the generalized Pareto distributions in R (Balkema and de Haan, 1974) and in R d (Rootzén and Tajvidi, 2006;Falk, Hüsler and Reiss, 2010). It leads to what we call generalized Pareto processes.
The paper is organized as follows: The Pareto processes will be dealt with in Section 2. As in the finite dimensional context it is convenient to study first generalized Pareto processes in a standardized form. This is done in Section 2.1. The general process is discussed in Section 2.2. In the meantime, in Section 2.1.1 is the discrete version of our approach to simple Pareto processes, leading to multivariate simple Pareto random vectors. Domain of attraction is discussed in Section 3.
In Section 4, we show that by using the stability property of generalized Pareto processes one can create extreme storm fields starting form independent and identically observations of storm fields.
In the following, operations like w 1 +w 2 or w 1 ∧w 2 with w 1 , w 2 ∈ C(S) mean respectively {w 1 (s)+w 2 (s)} s∈S and {w 1 (s)∧w 2 (s)} s∈S . Then with abuse of notation, operations like w+x or w∧x with w ∈ C(S) and x ∈ R mean respectively {w(s) + x} s∈S and {w(s) ∧ x} s∈S . Similarly for products and powers. Then e.g. we shall simply write 1 + γ X−bt at 1/γ In general, C(S) is the space of continuous real functions on S ⊂ R d equipped with the supremum norm.

The simple Pareto process
Let C + (S) be the space of non-negative real continuous functions on S, with S some compact subset of R d . We denote the Borel subsets of a metric space by B (·).
Theorem 2.1. Let W be a stochastic process in C + (S) and ω 0 a positive constant. The following three statements are equivalent: 1. (POT -peaks-over-threshold stability) (a) E (W (s)/ sup s∈S W (s)) > 0 for all s ∈ S, for all r > 1 and A ∈ B C + ω0 (S) , where rA means the set {rf, f ∈ A}, and , for all s ∈ S, for some Y and V = {V (s)} s∈S verifying: (a) V ∈ C + (S) is a stochastic process verifying sup s∈S V (s) = ω 0 a.s., and EV (s) > 0 for all s ∈ S, Definition 2.1. The process W characterized in Theorem 2.1, with threshold parameter ω 0 , is called simple Pareto process. The probability measure in (2.1) i.e., is called the spectral measure.
Some easy consequences of Theorem 2.1.3. are the following. The process W is stationary if and only if V is stationary. Independence at any two points s 1 , s 2 ∈ S, i.e. W (s 1 ) and W (s 2 ) being independent, is not possible. Complete dependence is equivalent to V ≡ ω 0 a.s. We shall come back to some of these issues.
Proof of Theorem 2.1. We start by proving that 1. implies 3. By compactness and continuity, sup s∈S W (s) < ∞ a.s. Take: Then (a), (b) and (c) are straightforward. Next we prove that 3. implies 2. Let for all r > 1 and B ∈ B C + ω0 (S) . Then, using in particular the independence of Y and V and P (sup s∈S W (s)/ω 0 > 1) = 1. Since P (rA) = r −1 P (A) holds for any of the above sets, it holds for all Borel sets in the statement. Finally, check that 2. implies 1. For any r > 1, by (c) and (a), Also for any B ∈ B C + ω0 (S) , since sup s∈S W (s) > ω 0 holds a.s. That is, it follows that sup s∈S W (s)/ω 0 is univariate Pareto distributed and, sup s∈S W (s) and W/ sup s∈S W (s) are independent.
The following properties are direct consequences: for all r > 1 and A ∈ B C + ω0 (S) .
2. (a) E (W (s)/ sup s∈S W (s)) > 0 for all s ∈ S, for all r > 1 and B ∈ B C + ω0 (S) . From (2.6) we see that the probability distribution of W serves in fact as the exponent measure in max-stable processes (cf. de Haan and Ferreira (2006), Section 9.3). Characterization 2. suggests ways for testing and modeling Pareto processes.
We proceed to express the distribution function of W in terms of the probability distribution of V from Theorem 2.1.3. and Definition 2.1.
Let w, W ∈ C + (S). Conditions W ≤ (>)w define the sets {f ∈ C + (S) : f (s) ≤ (>)w(s) for all s ∈ S} and (W ≤ w) defines the set {f ∈ C + (S) : f (s) > w(s) for at least one s ∈ S}. Hence note that W > w is not the complement of W ≤ w.
Take for the conditional expectation, defined in the usual sense and whenever ρ(B) = P (V ∈ B) > 0, with g a real functional (e.g. see Billingsley, 1995).
where the last but two equality follows by the fact that the second summand in the previous equality is zero and, from the independence of Y and V .
The following is obtained in the particular case of w being strictly positive: Let w, W ∈ C + (S), with w positive and W simple Pareto process. Then, Proof. (i) First consider the case inf s∈S w(s) ≥ ω 0 . Use Theorem 2.1, part 3., The probability measure of W on C + ω0 (S) can be extended to a measure ν on C + (S) while keeping the homogeneity relation (2.3) as follows: for any Borel set B such that This measure (the same as in (1.1)) is homogeneous of order -1: Then, the probability distribution of W is the restriction of ν to C + ω0 (S) i.e., for B ∈ B (C + (S)), Hence, by the homogeneity property of ν, (2.13) and (2.12) in that order: By (2.13), elementary set-measure operations and (2.14) in that order: The following formulas might also be useful: In particular, if P (V > 0) > 0 and sup s∈S w(s) > ω 0 , Proof. For (2.15), similarly to the proof of Proposition 2.1, For (2.17), using Y standard Pareto and independent of V . For c) note that Relation (2.19) indicates that one-dimensional marginals, conditional on the process being larger than ω 0 , behave like Pareto; similar observation is in Rootzén and Tajvidi (2006) in the context of lower-dimensional distributions.
Let s 1 , s 2 ∈ S and x > ω 0 . From (2.20), and, similarly That is, independence in the Pareto process between any two points is impossible. For later use we define next max-stable processes and give well known properties useful for our purposes. It is called simple if its marginal distributions are standard Fréchet, and then it will be denoted byη.
Proposition 2.3 (Penrose(1992), Theorem 5.). All simple max-stable processes can be generated in the following way. Consider a Poisson point process on Conversely each process with this representation is simple max-stable (and one can take V such that sup s∈R V (s) = c a.s. with c > 0).
The finite dimensional distributions ofη where computed in de Haan (1984). With the above representation for simple max-stable processes, for s 1 , . . . , s n ∈ R, x 1 , . . . , x n > 0, for all n ∈ N, (2.22) Note thatη is the maximum of infinitely many processes whereas W depends on just one of those processes (Theorem 2.1.3.).

The finite-dimensional setting
Simple Pareto random vectors can be constructed similarly as in Theorem 2.1 -Definition 2.1.
(2.23) for all r > 1 and B ∈ B D + ω0 with

25)
for all r > 1 and A ∈ B D + ω0 with Proof. Similar to the proof of Theorem 2.1.
From the finite dimensional representation it follows again that for having all marginals Pareto, one would have to have max(V (s 1 ), . . . , V (s n )) = ω 0 , for all s 1 , . . . , s n ∈ S and all integer n, which corresponds to V ≡ ω 0 a.s., i.e. the complete dependence case.
Nonetheless, we see that it is still possible that some finite dimensional marginals of a Pareto process are still Pareto. For example, consider a situation where the maximum of the process may only occur at some fixed locations in S and the related finite dimensional random vector includes these locations.
One can give formulas for distribution functions, following similar reasoning as before. The correspondent to Proposition 2.1 is given by, which corresponds to the one given in Rootzén and Tajvidi, in the correspondent region. Note that Rootzén and Tajvidi's formula (i.e. (2) in Definition 2.1) only holds for the vector x larger than the vector of the lower endpoints of the marginal distributions. The following example illustrates this. Consider the model related to two independent unit Fréchet. Then Rootzén and Tajvidi's formula corresponds to , (x > 0 and y ≥ 1) or (y > 0 and x ≥ 1) 0, 0 < x ≤ 1, 0 < y ≤ 1 but, it is undetermined otherwise. Then note that their formula is not able to capture the positive mass on the axis.

(2.30)
Another remark on Rootzén and Tajvidi (2006): their Theorem 2.2(ii) is not completely correct. It is not sufficient to require condition (6) of the same paper for x > 0. A counter example is given by and zero elsewhere. This distribution satisfies (6) for x > 0 but not for all x and it is not a generalized Pareto distribution. The generalized Pareto process W µ,σ,γ ∈ C(S) is defined by,

The generalized Pareto process
with all operations taken componentwise (recall the convention explained in the end of Section 1).
The result corresponding to Corollary 2.1 is, has standard Pareto distribution.
Related with stability or homogeneity properties we have: Proposition 2.4. For any generalized Pareto process W µ,σ,γ ,

Domain of attraction
The Next is an equivalent characterization of the domain of attraction condition, in terms of 'exceedances'.
For X a random elements of C(S), suppose the marginal distribution functions F s (x) = P (X(s) ≤ x) are continuous in x, for all s ∈ S. To simplify notation write the normalized process, for γ = {γ(s)} s∈S and normalizing functions a t = {a t (s)} s∈S > 0 and b t = {b t (s)} s∈S , all in C(S).

Condition 3.2.
For X ∈ C(S) suppose, for some a t > 0 and b t in C(S), uniformly in s, for all x > 1, (3.4) and

5)
for each B ∈ B C + 1 (S) with ρ(∂B) = 0, with ρ some probability measure on C + 1 (S). Note that this is the same as for max-stable processes; cf. Theorem 9.5.1 in de Haan and Ferreira (2006). Note also that (3.4)-(3.5) specify a simple Pareto process in the limit. Proof. Cf. Theorem 9.5.1 in de Haan and Ferreira (2006). The normalization there is 1 t(1 − F s (X(s))) s∈S , instead of (3.2) but the results are the same.
The following is a direct consequence: with A ∈ B C + 1 (S) , P (∂A) = 0 and W some simple Pareto process. The converse statement of Corollary 3.1 is as follows: Suppose that there exists a functionb u = {b u (s)} s∈S , that is continuous in s for each u and increasing in u, and with the property that P (X(s) >b u (s) for some s ∈ S) → 0 as u → ∞, and a continuous function (in s),ã u = {ã u (s)} s∈S > 0 such that, for some probability measureP on B (C(S)), for all A ∈ B (C(S)) andP (∂A) = 0. Then Conditions 3.1 and 3.2 are fulfilled.
Proof. By the conditions onb u , we can determine q = q(t) such that P (X(s) > b q(t) (s) for some s ∈ S) = 1/t. Then with b t (s) =b q(t) (s) and a t (s) =ã q(t) (s), for all C ∈ B (C(S)) andP (∂C) = 0. In particular, if inf{sup s∈S f (s) : f ∈ C} > 0 we have We proceed as usual in extreme value theory. Fix for the moment s ∈ S. It follows that for x > 0 Let U s be the inverse function of 1/P (X(s) > x) and V (s) be the inverse function of 1/P {f : f (s) > x}. Then It follows (Lemma 10.4.2, p.340, in de Haan and Ferreira (2006)) that for some real γ(s) and all x > 0 Since the limit process has continuous paths, the function γ must be continuous on S. Now replace t in (3.6) by ct where c > 0. Then and by (3.6) Write P (A) =P ((A γ − 1) /γ). Then  Example 3.2. Any Pareto process with spectral measure ρ is in the domain of attraction of a max-stable process where the underlying process V (cf. representation (2.21)) has probability measure ρ.
Hence, a regularly varying process such that (3.3) holds for the marginals, verifies Condition 3.2 with γ = 1/α, b t = t and a t = t/α; note that the index function is constant in this case.

View towards application and simulation 4.1 Towards application
The following remark may be useful for application. Suppose the domain of attraction condition (1.1) holds. Define B = {f ∈ C + (S) : f > 1}. Let A be a Borel set in C + (S). Then applying (1.1) twice we get This is the content of Corollary 3.1. It gives a limit probability distribution on B.
A similar reasoning holds with B replaced by a different set B ′ as long as which again is a probability distribution. Now we proceed as in the peaks-over-threshold method for scalar observations: let k = k(n) be a sequence of integers with lim n→∞ k(n) = ∞ and lim n→∞ k(n)/n = 0, as n → ∞. Suppose that we have n independent observations of the process X in the domain of attraction. Select those observations satisfying X(s i ) > b n/k (s i ), for i = 1, 2, . . . , p. The probability distribution of those selected observations is approximately the right-hand side of (4.1). This seems a useful applicable form of the peaks-over-threshold method in this framework as it suggests estimating the spectral measure using observations that exceed a threshold at some discrete points in the space only.

Towards simulation
'Deltares' is an advisory organization of the Dutch government concerning (among others) the safety of the coastal defenses against severe wind storms. One studies the impact of severe storms on the coast, storms that are so severe that they have never been observed. In order to see how these storms look like it is planned to simulate wind fields on and around the North Sea using certain climate models. These climate models simulate independent and identically distributed (i.i.d.) wind fields similar to the ones that could be observed (but that are only partially observed). Since the model runs during a limited time, some of the wind fields will be connected with storms of a certain severity but we do not expect to see really disastrous storms that could endanger the coastal defenses. The question put forward by Deltares is: can we get an idea how the really disastrous wind fields look like on the basis of the 'observed' wind fields? We want to show that this can be done using the generalized Pareto process.
Consider a continuous stochastic processes {X(s)} s∈S where S is a compact subset of R d . Suppose that the probability distribution of the process is in the domain of attraction of some max-stable process i.e., there exist functions a n (s) > 0 and b n (s) (s ∈ S) such that the sequence of i.i.d. processes max 1≤i≤n X i (s) − b n (s) a n (s) s∈S converges to a continuous process, say η, in distribution in C(S). Then η is a max-stable process. Define As before, suppress the s from now on. Then, with t 0 some large constant, Since, a tt0 (s)t −γ(s) 0 a t (s) uniformly for s ∈ S, the limit of this probability, as t → ∞, is the same as the limit of which is P (W ∈ A) by Corollary 3.1.
In this subsection we are not so much interested in estimating the joint limit distribution (which is the peaks-over-threshold method) but in the fact that the two conditional distributions (4.2) and (4.3) are approximately the same.
Suppose for example that we have observed wind fields over a certain area during some time. Then we are likely to find some rather heavy storms i.e. ones that satisfy X ≤ b n . These are the moderately heavy storms. However we want to know how the storm field of a really heavy storm (i.e. X ≤ b N with N > n) looks like. That is exactly what relation (4.2) does. Take a moderately heavy storm X and transform it to T ← n N n T n X. This results in the storm field of a really heavy storm by relation (4.2).
Notice then what we do here is similar to prediction or kriging, not estimating a distribution function.
The reasoning above also holds with estimated functions of γ, a and b, on the basis of k-th upper order statistics and taking t = n/k.  Figure 1: (a) The processesT n/k X i for whichR T n/k Xi > 1 obtained from moving maximum process with standard Gaussian density; (b) the corresponding lifted processesT ← n/k t 0Tn/k X i with t 0 = 10

Simulations
We exemplify the lifting procedure with the process X(s) = Z(s) γ(s) , with γ(s) = 1 − s(1 − s) 2 , s ∈ [0, 1], and Z is the moving maximum process with standard Gaussian density. The Z process can be easily simulated in R-package due to Ribatet (2011). In Figures 1-2 are represented the 11 out of 20 of these processes, normalized and for whichR T n/k Xi > 1, and the corresponding lifted processesT ← n/k t 0Tn/k X i with t 0 = 10.  Figure 2: All processes:T n/k X i -slashed lines -andT ← n/k t 0Tn/k X i -continuous lines for whichR T n/k Xi > 1