Spatial extreme value theory and, especially, max-stable processes are widely applied tools to assess risks in environmental science. These processes are motivated by the study of
where are independent observations of a sample-continuous process , modeling a phenomenon of interest such as rainfall or temperature on some region . The scaling functions and , , are both continuous in . Functional limits obtained from this construction as , named the class of max-stable processes, are appealing models for spatial extremes. Their realizations, however, are composed of different single events , which prohibits direct interpretation and renders efficient inference and simulation challenging (e.g., Dombry et al., 2016; Thibaud et al., 2016).
It is often more natural to study threshold exceedances, or, more precisely, the extremal behavior of , , where is a functional on the space of continuous functions on . Buishand et al. (2008), for instance, consider the daily rainfall over a certain region , and therefore choose . Using the same functional, Coles and Tawn (1996) relate the tail of the distribution of the integral to the tail of the distribution at a single location, and Ferreira et al. (2012) formalize this idea through the so-called reduction factor. For general homogeneous functionals , Dombry and Ribatet (2015) characterize the functional limits of threshold exceedances , for a high threshold .
In this paper we follow the approach of Coles and Tawn (1996) and Ferreira et al. (2012) in order to investigate the tail behavior of more general functionals . Under certain conditions we show that, for any ,
for sufficiently large . This means that the tail of the -functional of behaves like the tail at an individual location times a reduction factor , which we call the -extremal coefficient. In different contexts, the interpretation of this coefficient might differ, but intuitively summarizes the effect of spatial extremal dependence in on the risk diversification through the functional .
-extremal coefficient relates the tail of the univariate random variableto the multivariate or spatial extremal dependence in
. A major advantage of this functional perspective is that it produces return level estimates that are consistent with respect to the underlying structure of, even when considering different aggregation functionals applied to the same process . Indeed, for functionals , we study the multivariate tail behavior of , which turns out to be in the max-domain of attraction of a multivariate max-stable distribution.
Popular models for the functional limit of the maxima in (1) are Brown–Resnick processes that take a similar role in spatial extremes as Gaussian processes in classical geostatistics. The reason for this is that the former are essentially the only such limits when is a stationary Gaussian process and an additional rescaling is allowed (Kabluchko et al., 2009). This connection can be exploited to perform efficient inference (Wadsworth and Tawn, 2014; Engelke et al., 2015; Thibaud et al., 2016) and simulation (Dombry et al., 2013, 2016; Oesting and Strokorb, 2017)
for Brown–Resnick processes based on densities and sampling algorithms of Gaussian random vectors. In our framework, this link to Gaussian distributions allows us to use results from the geostatistical literature on data aggregation(e.g., Wackernagel, 2003) to obtain explicit expressions for and the extremal dependence in if the limiting process in (1) is a Brown–Resnick process with Gumbel margins.
An important consequence of our findings is that they allow, under certain assumptions, to recover the tail distribution of based only on information from the aggregated vector. This is similar to inferring the extremal dependence of based only on extremal coefficients (cf., Schlather and Tawn, 2003). In meteorology, for instance, large scale climate models provide only data over grid cells, but practical questions require risk assessment at point locations such as cities or other infrastructural sites. Techniques to perform this transition from large to small scales are summarized under the notion of downscaling. In the second part of the paper we thus propose a statistical downscaling method to infer in a spatially consistent way the tail behavior of the underlying stochastic process
based on the observed extremes of the aggregated data. Relevant outputs will be the exceedance probabilities at point locations and simulations of spatial extreme events of, both unconditionally and conditionally on the observed aggregated extremes. We apply this procedure to coarse scale gridded temperature data in the south of France from the e-obs data set (Haylock et al., 2008). The fitted model provides, for instance, fine-resolution simulations of the warmest day during the heatwave conditionally on the observed grid values.
2 Limit results for extremes of aggregated data
2.1 Background on extremes
Let be a compact subset of a complete separable metric space. By we denote the space of real-valued functions on equipped with the supremum norm , defined by , and the corresponding Borel -algebra .
We consider a sample-continuous stochastic process , which we assume to be in the max-domain of attraction of a max-stable process with common marginal extreme value index . More precisely, for independent copies of , there exist functions , , both continuous in , such that as , the process of componentwise maxima defined in (1) converges in distribution on the space , i.e.,
where denotes the law of a process . By definition, the process in the limit is max-stable, and it is simple in the sense that it is normalized to have unit Fréchet margins (cf., de Haan and Ferreira, 2006, Chapter 9). Moreover, for any , the margin then is in the max-domain of attraction of an extreme value distribution
for all . The different distributions are called -Fréchet for , Gumbel for and -Weibull for , respectively. The assumption of a spatially constant in (3) is common in the literature since it is required to obtain meaningful theoretical results, and it is usually a reasonable hypothesis in applications.
where are the points of a Poisson point process on with intensity measure and the spectral functions , , are independent copies of some non negative sample-continuous process with for all .
In the sequel we assume that is non negative for the Fréchet case , while for the Weibull case each , , is assumed to have the same upper endpoint . Finally, we provide the example of the widely used class of Brown–Resnick max-stable processes.
Let be a centered Gaussian process with variogram . A Brown–Resnick process is the max-stable process in (5) where the spectral functions follow the distribution of
The distribution of only depends on the variogram , and for , the finite dimensional distribution of is called the Hüsler–Reiss distribution (Hüsler and Reiss, 1989) with parameter matrix ; more details can be found in Appendix B and in Brown and Resnick (1977), Kabluchko et al. (2009) and Kabluchko (2011).
2.2 Univariate limiting distributions of aggregated data
We first derive the univariate asymptotic distribution of aggregated data. Following Ferreira et al. (2012), we assume that the normalizing functions can be decomposed asymptotically into positive functions and in the sense that
In the Gumbel case , the left-hand side of (6) is even assumed to be equal to zero, i.e.,
For data aggregation, we consider a positively homogeneous functional , i.e., satisfies for all , . We further assume that is uniformly continuous, and we use the notation and interchangeably.
Let be a positively homogeneous and uniformly continuous functional on . Further, assume that (3) and (6) hold. If , the spectral functions belonging to the process in (3) are assumed to be strictly positive. Then, for , we have
For , we further require that (7) holds and that is linear. In this case,
Theorem 1 is formulated for threshold exceedances, but, using well-known equivalences from univariate extreme value theory, it could be easily reformulated to describe the limiting behavior of , where are independent copies of .
We call the quantity the -extremal coefficient since it describes the change of the upper tail of the -aggregated data compared to the tail of the univariate marginal data. Our definition of in Theorem 1 contains a normalization by , making it invariant under multiplication of by a constant and thus simplifying interpretation. Indeed, for and , we observe that
The interpretation of this coefficient might differ depending on the respective context and risk functional . In general, summarizes the effect of the spatial extremal dependence in on the diversification of the risk through functional . Importantly, not only the dependence but also the marginal tail index may effect the coefficient , which we stress in Theorem 1 and henceforth by the index .
The concept of the -extremal coefficient extends and unifies various notions in extreme value statistics and applied sciences such as extremal coefficients, diversification factors in portfolios and areal reduction factors. We present these and other examples for illustration, always assuming that satisfies the conditions of Theorem 1.
The important case where is a compact region and
was first studied in
Coles and Tawn (1996) and Buishand et al. (2008) in the framework of total areal rainfall, and
then formalized by Ferreira et al. (2012). In this case of a spatial average, the coefficient
is popular in environmental science
where it is called the areal reduction factor. Hydrologists use it to convert
quantiles of point rainfall to quantiles of total rainfall over a river catchment
of interest. Interestingly, this coefficient satisfies
is popular in environmental science where it is called the areal reduction factor. Hydrologists use it to convert quantiles of point rainfall to quantiles of total rainfall over a river catchment of interest. Interestingly, this coefficient satisfiesfor , and for (Ferreira et al., 2012, Prop. 2.2). That means that average rainfall is less extreme than point rainfall if the marginals have finite expectation, as typically encountered in practice, and more extreme if they have infinite expectation.
If is a finite set and is a weighted sum with fixed , then Zhou (2010) and Mainik and Embrechts (2013) computed the corresponding coefficient for . In this setup, , , are interpreted as dependent, heavy-tailed risk factors, and represents the diversification in the portfolio . More precisely, the value at risk of for high levels can be expressed as the value at risk of a single factor times a constant that involves the -extremal coefficient . Theorem 1 yields an analogous result also for light tailed risk factors.
Another well-known example is the case of being a finite set and . If we further have , then corresponds to the classical extremal coefficient (Schlather and Tawn, 2003), a number between and that is usually interpreted as the number of asymptotically independent random variables among . A similar interpretation is valid also if is an arbitrary compact subset, and is a spatial extension of the classical extremal coefficient.
As a last example, we consider energy functionals of the type that appear in various applications in physics. In the case of being a wind field, is proportional to the integrated kinetic energy over a region , which is an indicator for the potential damage caused by the corresponding storm event (eg. Powell and Reinhold, 2007).
The expressions (9) and (11) for the -extremal coefficient are expected values of functions of the spectral process . The distribution of the latter is known for most popular models, and it includes truncated Gaussian processes (Schlather, 2002; Opitz, 2013) and -Gaussian processes (Brown and Resnick, 1977; Kabluchko et al., 2009), for instance. Numerical evaluation of is thus readily implemented through simulations of . In the important case of and corresponding to a log-Gaussian process, we obtain a closed form expression for .
Suppose that and is a Brown–Resnick process on a compact set , as introduced in Example 1. The extremal coefficient of the spatial average then is
Let and , , and consider the popular power variogram model, namely for some . In this case we obtain
2.3 Multivariate limiting distributions of aggregated data
In the previous section we derived the univariate tail distribution of data aggregated through a functional . In applications we often observe data through several different functionals, e.g., the integrals over not necessarily disjoint areas. The consistency of return level estimates discussed in the introduction has even more important implications when different risk functionals are applied to the data. The univariate tail of each aggregation could be estimated separately, but the dependence between the tails would not be captured. We consider therefore arbitrary positively homogeneous, uniformly continuous functionals , and we aim at describing the multivariate tail behavior of the vector .
The proof of the following theorem can be found in Appendix A.
Let be positively homogeneous and uniformly continuous functionals on . Further, assume that (3) and (6) hold. If , the spectral functions belonging to the process in (3) are assumed to be strictly positive. Then, for and
For , we further require that (7) holds and that the functionals , , are linear. In this case, for any ,
The above theorem states that the vector of aggregations is in the max-domain of attraction of the multivariate max-stable distribution with exponent measure given by the right-hand side of (13) or (14), respectively. For the th margin, for , the scale of the Weibull or Fréchet distribution is , and for the location parameter of the Gumbel distribution is . This recovers the univariate results in Theorem 1. For details on multivariate domains of attraction and exponent measures, see Resnick (2008, Chapter 5). In general this max-stable distribution will not be available in closed form, but for the purpose of evaluating risk regions for , the exponent measure can be approximated by Monte Carlo methods. In the following important special case, we can compute the multivariate distribution explicitly.
Consider the same framework as in Example 6, namely compact, and is in the max-domain of attraction of Brown–Resnick process with spectral functions . Suppose that for all , the functional is the spatial average over the compact region , respectively. Since is log-Gaussian in this case, the random vector
is multivariate Gaussian, and its variogram matrix can be computed explicitly; see Appendix B. The exponent measure in (14) therefore corresponds to a -variate Hüsler–Reiss distribution with dependence matrix whose th margin has a Gumbel distribution with location parameter given in (12).
3 Statistical Inference
We suppose that we observe independent data , , from the process , but only through the aggregation functionals satisfying the conditions from Theorem 2. The observations are therefore -dimensional and of the form
Making use of the limit results in Theorems 1 and 2, we aim to infer the extremal behavior of the whole process from the observed aggregated data. This requires estimation of both the marginal tail behavior and the extremal dependence of . Naturally, further assumptions are needed to render this problem well-defined.
We suppose that the process is in the functional max-domain of attraction of a max-stable process as in (3) with marginal distributions of of the form (4) for all . A natural and fairly general assumption on the marginal distributions of is to belong to a location-scale family, i.e., for some distribution function and continuous , ,
for any fixed . Since is in the max-domain of attraction of , the distribution of must converge to as . In particular, must satisfy for all with and appropriate functions and . This implies that the normalizing functions and of can be chosen as
Moreover, if , without loss of generality, we may assume by the same arguments as in the proof of Theorem 2.
We impose a parametric structure on the marginal scale and location parameters, i.e., the unknown functions and , respectively, and the extremal dependence of , which is given by the exponent measure of . For the marginal distributions, we assume that and belong to parametric families of functions and where and are appropriate subsets of and , respectively. For the dependence, we suppose that the probability measure induced by the spectral function of the limiting max-stable process belongs to a parametric class with . Further, the joint normalization constants and need to be estimated for some large .
3.2 Least squares fit based on marginal estimates
Throughout the rest of this section, for the sake of simplicity we assume that is known. Estimation for the case that is unknown can be performed analogously, see Appendix C.
As a first approach, we approximate the tail of the distribution of separately for each . From (14), we obtain for exceedances over sufficiently large and ,
or, equivalently, for block maxima,
where the location parameters and the scale parameters , , are given by
respectively. While the asymptotic behavior of and is uniquely determined by Equation (17), additional assumptions on and , such as and , are necessary to ensure the identifiability of , , , and from Equations (19) and (20).
For large , estimates and for and can be obtained using well-known techniques from univariate extreme value statistics based on peaks-over-threshold or block maxima approaches, for instance. Here, the value of is typically closely related to the choice of the threshold and the block size, respectively; see Appendix C for details. Based on these estimates, we define a weighted least squares estimator
are the variances of the estimatorsand .
3.3 Censored likelihood for the joint tail behavior
Alternatively, we can estimate making use of the multivariate tail behavior of the whole vector . For sufficiently large and , by Theorem 2,
is the exponent measure of a max-stable vector with standard Gumbel margins. Thus, the parameter vector can be estimated by a censored likelihood approach. Define a vector whose th element is a suitably high marginal threshold for , such as its empirical -quantile, and let . Denoting the normalized thresholds and data by and with
respectively, we let be the of the log-likelihood
where and are the partial derivatives of in directions . By the homogeneity of , it can be seen that the likelihood (22) asymptotically does not depend on the specific choice of , but only on the . This likelihood corresponds to multivariate threshold exceedances and their approximation by Pareto processes (cf., Thibaud and Opitz, 2015). The censoring of the exponent measure reduces possible bias for observations below the marginal threshold that might not yet have converged to the limit model; see Wadsworth and Tawn (2014).
4 Simulation of Extreme Events
Environmental risk assessment is often based on rare event simulation of scenarios with long return periods. Two kinds of simulations are typically required: unconditional simulations of a given or fitted model capturing the spatial extent and the variability of possible extreme events; and simulations at points of interest conditional on a particular event that was only observed at different locations or scales. Conditional and unconditional simulations have for instance been studied for max-stable processes (Dombry et al., 2013, 2016) and for threshold exceedances (Thibaud and Opitz, 2015; de Fondeville and Davison, 2017).
In this section, we discuss how the multivariate result in Theorem 2 allows us to perform these two kinds of simulations for extreme events of the process . We assume that the process satisfies the assumptions of Theorem 2 for known normalizing functions and with representation (16), extreme value index , and known distribution of the spectral process . For simplicity, we again restrict to the case , but the procedure can be adapted for the case .
In order to simulate at a finite number of locations , we artificially augment the vector of functionals to , where is the point evaluation at location , . We apply Theorem 2 to this augmented vector to obtain
and and , , , are defined as in Equations (19) and (20), respectively. In other words, is in the-max-domain of attraction of a max-stable distribution with standard Gumbel margins and spectral vector .
In the framework of conditional simulation of an extreme event, the aggregated data , are observed and one of them, say , is assumed to be large. Reformulating Theorem 2 in terms of threshold exceedances, we obtain the convergence in distribution
as , where is a standard exponential random variable and, independently of , is a -dimensional random vector with the transformed distribution given in Dombry et al. (2016, Proposition 1). For most popular models in spatial extremes, can be simulated easily. Using approximation (23) with for some large , we can perform conditional simulation of the vector in the following way.
Simulate a realization of the conditional distribution of given that , .
As a conditional realization of , return with
For unconditional simulation, one is typically interested in extreme events in the sense that at least one of the functionals exceeds a high threshold. Therefore, we replace the conditioning event in (23) by , such that the vector in the limiting law in (23) becomes a vector that is normalized with respect to the maximum of its first components (cf., Dombry and Ribatet, 2015). Noting that can be generated by rejection sampling (cf., de Fondeville and Davison, 2017), we can adapt the conditional simulation procedure to obtain an unconditional extreme sample. Indeed, it suffices to let in (i) be a realization of the unconditional distribution of , and to replace the constant in Equation (24
) in (ii) by a realization of the standard exponential distribution.
In order to perform conditional and unconditional simulation, the multivariate tail behavior of the vector in the sense of Theorem 2 is required. For our running example of a limiting Brown–Resnick process, the following makes this explicit.
As in Example 7, let be compact, and is in the max-domain of attraction of a Brown–Resnick process. The aggregation functionals are spatial averages over compact regions , , or point evaluations at locations , . The vector then satisfies the assumptions of Theorem 2, and it is in the max-domain of attraction of a multivariate Hüsler–Reiss distribution with dependence matrix
The entries of the four sub-matrices and the explicit form of the exponent measure are given in Appendix B. In this case, the above algorithms essentially reduce to conditional and unconditional simulation of Gaussian processes.
5 Application: downscaling extremes
5.1 Statistical downscaling
Environmental data can be classified into two broad categories. On the one hand, station measurements are obtained through direct observation of the physical quantity. This type of data refers to a precise location in space, but it may suffer from inhomogeneities between stations due to varying record lengths and differences between measurement instruments, and, moreover, it usually has a sparse spatial coverage. Gridded databases, for instance generated by climate models, on the other hand, cover a large region or even the entire globe, but at a coarse scale where data points can be considered as an aggregation of the physical variable.
Understanding the link from these gridded data to point measurements is an important area of research in environmental sciences called downscaling. Besides dynamical downscaling procedures based on the solution of partial differential equations describing the physical processes, a large number of downscaling techniques relying on the statistical relationship between variables at different scales have been applied. Most of these techniques focus on central characteristics of the distribution such as mean and variance. In geostatistics, for instance, the so-called change of support has been extensively studied for Gaussian processes(cf., Chiles and Delfiner, 2012, and references therein). There are only few examples of statistical downscaling procedures for extremes. Mannshardt-Shamseldin et al. (2010) and Kallache et al. (2011) follow an approach related to univariate extreme value theory, and Bechler et al. (2015) and Oesting et al. (2017) propose conditional simulation from a spatial max-stable process that has been estimated from station measurements.
Here, using the theoretical results in Section 2, we extend the idea of changing the support of a stochastic process to the context of extremes, basing inference only on aggregated observations . These might come from gridded data sets, as in our case, supposing that the grid values represent an aggregation of the underlying physical quantity. If additional station measurements are available, they can also be used. Outputs of the method will be return level estimates at point locations, as well as unconditional and conditional simulations of rare events in the region . The method allows for the estimation of marginal characteristics such as return levels at point locations, as well as unconditional and conditional simulations of rare events on the entire region .
5.2 Application to extreme temperature in the South of France
We apply our downscaling procedure to daily temperature maxima in Europe from the e-obs data set (Haylock et al., 2008), which covers the period from to with a grid resolution. To avoid potential temporal non-stationarity, we restrict the study to the summer period, i.e., July and August. Our study region is a subset of the gridded product located in the south of France, in the west of Perpignan; see Figure 1. The region is mountainous and thus altitude appears to be a natural covariate for our model. The underlying spatial process of temperatures is denoted by , and the observations on day can be considered as the spatial averages over the cells in , where , and is the number of days in the given time span of
years. The null hypothesis that the marginal tails of the aggregated data are in the Gumbel domain of attraction cannot be rejected, and we thus assume in sequel that.
Throughout we assume the same setting as in Section 3.1, namely that the marginal distributions of belong to a location-scale family for all parameterized through the functions
where , and denote the altitude, longitude and latitude at location , respectively. We further suppose that is in the functional max-domain of attraction of a max-stable process belonging to a parametric family , for which we consider the Brown–Resnick processes introduced in Example 1, parameterized by for the anisotropic power variogram
with and anisotropy matrix
In Sections 3.2 and 3.3 we discussed two approaches to estimate the parameters of this model, namely least squares estimation based on univariate location and scale estimates, and censored likelihood estimation for multivariate threshold exceedances. The formulas required for the implementation of these approaches have been derived in Sections 2 and 3 and in Appendix B. For least squares estimation, this includes the explicit expression (12) for the univariate -extremal coefficient. For censored likelihood estimation of the model parameters in (22), we require the partial derivatives of the exponent measure , which can be obtained as in Asadi et al. (2015, Section 4.3.2); see Appendix B for more details. In order to assess the effectiveness and to compare the efficiency of the two methods, in Appendix D we perform a simulation study with a setup similar to this application. It turns out that the censored likelihood approach is significantly more efficient since it uses the full information on extremal dependence.
The parameters of our model for temperature extremes are therefore fitted using the censored likelihood procedure based on all observations where at least one component exceeds its respective empirical quantile. To avoid possible temporal dependence we keep only observations that are at least days apart, yielding a set of events. The parameter estimates are displayed in Table 1
where standard deviations are obtained using a jackknife procedure withblocks of size ; censored maximum likelihood is performed repeatedly with one block left out.
We assess the model fit in the diagnostic plots shown in Appendix E. First, we check the marginal distributions implied by the fitted linear model by comparing them in quantile-quantile plots to the observations; see Figure A1. The model provides a good fit for most stations and the quantiles of the fitted model generally remain in the confidence bounds obtained by parametric bootstrap. For a small number of stations, the model slightly over-estimates return levels.
Verification of the dependence structure is based on a graphical comparison of the pairwise extremogram (Davis and Mikosch, 2009) from the fitted multivariate Hüsler–Reiss model to its empirical counterpart based on the gridded observations. The extremogram values were significantly larger than zero for increasing thresholds and stable around the empirical quantile, validating the asymptotic dependence model. Figure A2 shows that the fitted variogram model successfully captures the major trend of the cloud of points. The effect of spatial anisotropy seems to be rather weak, which is also reflected in the parameter estimate for close to .
The fitted marginal model allows us to obtain return level maps for point locations at arbitrarily fine resolutions. In Figure 2, we produced such maps for the and year return periods. The full fitted model of marginal distributions and dependence structure further enables us to conditionally and unconditionally generate spatial extreme events of temperature fields at both a coarse and a fine resolution grid via the simulation procedures described in Section 4. Figure 3, for instance, displays two high resolution simulations of the temperature field conditionally on the observed aggregated temperatures during the warmest day of the heatwave. The simulations show that extreme temperatures at fine resolutions can be remarkably larger than at a coarse scale. Moreover, both simulations are constrained to have the same observed averages on the grid boxes, but they may exhibit different spatial patterns. This illustrates the variability of such a heatwave and provides practitioners with a set of possible scenarios that can be used for risk assessment.
We would like to thank Anthony C. Davison for helpful comments and discussions, and the Swiss National Science Foundation for financial support.
- Asadi et al.  P. Asadi, A. C. Davison, and S. Engelke. Extremes on river networks. Ann. Appl. Stat., 9(1):2023–2050, 2015.
- Bechler et al.  A. Bechler, M. Vrac, and L. Bel. A spatial hybrid approach for downscaling of extreme precipitation fields. J. Geophys. Res. Atmos., 120(10):4534–4550, 2015.
- Brown and Resnick  B. M. Brown and S. I. Resnick. Extreme values of independent stochastic processes. J. Appl. Probab., 14(4):732–739, 1977.
- Buishand et al.  T. A. Buishand, L. de Haan, and C. Zhou. On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat., 2(2):624–642, 2008.
- Chiles and Delfiner  J.-P. Chiles and P. Delfiner. Geostatistics: Modeling Spatial Uncertainty. John Wiley & Sons, 2012.
- Coles and Tawn  S. G. Coles and J. A. Tawn. Modelling extremes of the areal rainfall process. J. R. Stat. Soc. Ser. B Stat. Methodol., 58(2):329–347, 1996.
- Davis and Mikosch  R. A. Davis and T. Mikosch. The extremogram: a correlogram for extreme events. Bernoulli, 15:977–1009, 2009.
- de Fondeville and Davison  R. de Fondeville and A. C. Davison. High-dimensional peaks-over-threshold inference. Available from https://arxiv.org/abs/1605.08558, 2017.
- de Haan  L. de Haan. A spectral representation for max-stable processes. Ann. Probab., 12(4):1194–1204, 1984.
- de Haan and Ferreira  L. de Haan and A. Ferreira. Extreme Value Theory: An Introduction. Springer, Berlin, 2006.
- Dombry and Ribatet  C. Dombry and M. Ribatet. Functionnal regular variations, Pareto processes and peaks over threshold. Stat. Interface., 8(1):9–15, 2015.
- Dombry et al.  C. Dombry, F. Eyi-Minko, and M. Ribatet. Conditional simulation of max-stable processes. Biometrika, 100(1):111–124, 2013.
- Dombry et al.  C. Dombry, S. Engelke, and M. Oesting. Exact simulation of max-stable processes. Biometrika, 103(2):303–317, 2016.
- Engelke et al.  S. Engelke, A. Malinowski, Z. Kabluchko, and M. Schlather. Estimation of Hüsler–Reiss distributions and Brown–Resnick processes. J. R. Stat. Soc. Ser. B Stat. Methodol., 77(1):239–265, 2015.
- Ferreira et al.  A. Ferreira, L. de Haan, and C. Zhou. Exceedance probability of the integral of a stochastic process. J. Multivar. Anal., 105(1):241–257, 2012.
- Giné et al.  E. Giné, M. Hahn, and P. Vatan. Max-infinitely divisible and max-stable sample continuous processes. Probab. Th. Rel. Fields, 87(2):139–165, 1990.
- Haylock et al.  M. R. Haylock, N. Hofstra, A. M. G. Klein Tank, E. J. Klok, P. D. Jones, and M. New. A european daily high-resolution gridded data set of surface temperature and precipitation for 1950–-2006. J. Geophys. Res. Atmos., 113(D20), 2008.
- Hüsler and Reiss  J. Hüsler and R.-D. Reiss. Maxima of normal random vectors: between independence and complete dependence. Statist. Probab. Lett., 7(4):283–286, 1989.
- Kabluchko  Z. Kabluchko. Extremes of independent Gaussian processes. Extremes, 14(3):285–310, 2011.
- Kabluchko et al.  Z. Kabluchko, M. Schlather, and L. de Haan. Stationary max-stable fields associated to negative definite functions. Ann. Probab., 37(5):2042–2065, 2009.
- Kallache et al.  M. Kallache, M. Vrac, P. Naveau, and P.-A. Michelangeli. Nonstationary probabilistic downscaling of extreme precipitation. J. Geophys. Res.-Atmos., 116, 2011.
- Mainik and Embrechts  G. Mainik and P. Embrechts. Diversification in heavy-tailed portfolios: properties and pitfalls. Annals of Actuarial Science, 7(1):26–45, 2013.
- Mannshardt-Shamseldin et al.  E. C. Mannshardt-Shamseldin, R. L. Smith, S. R. Sain, L. O. Mearns, and D. Cooley. Downscaling extremes: a comparison of extreme value distributions in point-source and gridded precipitation data. Ann. Appl. Stat., 4:484–502, 2010.
- Oesting and Strokorb  M. Oesting and K. Strokorb. Efficient simulation of Brown–Resnick processes based on variance reduction of Gaussian processes. Available from https://arxiv.org/abs/1709.06037, 2017.
- Oesting et al.  M. Oesting, L. Bel, and C. Lantuéjoul. Sampling from a max-stable process conditional on a homogeneous functional with an application for downscaling climate data. Scand. J. Stat., 2017. To appear.
- Opitz  T. Opitz. Extremal processes: elliptical domain of attraction and a spectral representation. J. Multivariate Anal., 122(1):409–413, 2013.
- Penrose  M. D. Penrose. Semi-min-stable processes. Ann. Probab., 20(3):1450–1463, 1992.
- Powell and Reinhold  M. D. Powell and T. A. Reinhold. Tropical cyclone destructive potential by integrated kinetic energy. Bull. Am. Meteorol. Soc., 88(4):513–526, 2007.
- Resnick  S. I. Resnick. Extreme Values, Regular Variation and Point Processes. Springer, New York, 2008.
- Schlather  M. Schlather. Models for stationary max-stable random fields. Extremes, 5(1):33–44, 2002.
- Schlather and Tawn  M. Schlather and J. A. Tawn. A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika, 90(1):139–156, 2003.
- Thibaud and Opitz  E. Thibaud and T. Opitz. Efficient inference and simulation for elliptical Pareto processes. Biometrika, 102(2):855–870, 2015.
- Thibaud et al.  E. Thibaud, J. Aalto, D. S. Cooley, A. C. Davison, and J. Heikkinen. Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures. Ann. Appl. Stat., 10(4):2303–2324, 2016.
- Wackernagel  H. Wackernagel. Multivariate Geostatistics. Springer, New York, 2003.
- Wadsworth and Tawn  J. L. Wadsworth and J. A. Tawn. Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika, 101(1):1–15, 2014.
- Zhou  C. Zhou. Dependence structure of risk factors and diversification effects. Insurance Math. Econom., 46(3):531–540, 2010.
Appendix A Proof of Theorem 2
Condition (3) implies that the exponent measure of , defined by
where and denote the analogues to and for non negative functions, verifies
Closely related to the spectral process , the measure