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STS430 Eustasio D.B. et al.

⁄
⁄
′

[ℒ [{ , ()} 1 ] , ℒ [{ ′ , ()} 1 ]] ≤ 1 ∑ ( , ).

=1
Hence, the Wasserstein distance of the variance of two collections of
distributions can be controlled using the distance between the distributions.
The main consequence of this fact is that the minimal alignment cost can also
be bootstrapped as soon as a distributional limit theorem exists for , (), as
in the discussion above. In Sections ?? and ?? below, we present distributional
results of this type in the one-dimensional case. We note that, while general
Central Limit Theorems for the empirical transportation cost are not available
in dimension > 1, some recent progress has been made in this direction;
see, e.g., [30] for Gaussian distributions and [32], which gives such results for

distributions on ℝ with finite support. Further advances along these lines
would make it possible to extend the results in the following section to higher
dimensions.

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