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Let P 2,ac be the set of Borel probabilities on R d with finite second moment and absolutely continuous with respect to Lebesgue measure. We consider the problem of finding the barycenter (or Fréchet mean) of a finite set of probabilities ν 1 , . . . , ν k ∈ P 2,ac with respect to the L 2 −Wasserstein metric. For this task we introduce an operator on P 2,ac related to the optimal transport maps pushing forward any µ ∈ P 2,ac to ν 1 , . . . , ν k . Under very general conditions we prove that the barycenter must be a fixed point for this operator and introduce an iterative procedure which consistently approximates the barycenter. The procedure allows effective computation of barycenters in any location-scatter family, including the Gaussian case. In such cases the barycenter must belong to the family, thus it is characterized by its mean and covariance matrix. While its mean is just the weighted mean of the means of the probabilities, the covariance matrix is characterized in terms of their covariance matrices Σ 1 , . . . , Σ k through a nonlinear matrix equation. The performance of the iterative procedure in this case is illustrated through numerical simulations, which show fast convergence towards the barycenter.

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1 Although i.i.d.-ness can be relaxed into exchangeability, we are sticking to the former. 2 Typical examples are linear models, with Z i (θ θ θ) = X i − c i θ θ θ (c i a q-vector of covariates and θ θ θ ∈ R q ), or firstorder autoregressive models, with Z i (θ ) = X i − θX i−1 (where i denotes time and θ ∈ (−1, 1); see, for example, Hallin and Werker (1999)), etc.3 Those ranks indeed are maximal invariant under the group of continuous monotone increasing transformations of Z 1 (θ θ θ 0 ), . . . , Z n (θ θ θ 0 ); see, for instance, Example 7 in Lehmann and Scholz (1992).

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The Cramér-Wold theorem states that a Borel probability measure P on R d is uniquely determined by its one-dimensional projections. We prove a sharp form of this result, addressing the problem of how large a subset of these projections is really needed to determine P. We also consider extensions of our results to measures on a separable Hilbert space. As an application of these ideas, we derive a simple, universally consistent goodness-of fittest for data taking values in a Hilbert space.

The assessment of geoheritage is a crucial task for the establishment of ranks according to their value, an important step for the elaboration of catalogues and the implementation of protection and use programmes. One important problem that permeates the different stages of geoheritage assessment is subjectivity in the selection and application of criteria. Evaluation methods frequently used are based on the assignment of values through the expert's judgement (direct methods). In the last few decades, parametric methods, based on the measurement of specific features of geomorphosites have been increasingly applied. This type of procedure allows different operators to obtain similar results if certain criteria are accepted, but it requires more time and effort. A parametric method is proposed here based on three sets of criteria: intrinsic quality, potential for use and protection needs. The method has been tested on a series of coastal geomorphosites. Specific criteria, parameters to express them and value ranks are proposed. A statistical approach has been applied to the criteria to identify the most significant ones and their relative weights. The analysis enabled reduction of the number of criteria, eliminating redundancies. The new set of criteria was applied to assess the value of geomorphosites located in the north of Spain. The classification was compared with the ones obtained by both the parametric method using a larger number of parameters and a direct evaluation applied to the same sites. Results show a quite good coincidence. It is concluded that this approach makes it possible to obtain reliable classifications of geomorphosites based on clearly defined criteria and replicable procedures, but with a much lower effort, using a limited number of criteria.

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