Wasserstein metrics and weak convergence

For p at least one, let Pₚ be probability measures on a complete separable metric space with finite integral of distance from a reference point raised to p. The choice of reference point does not change this class. Define Wₚ by minimizing the expected p-th power of distance over couplings and taking the p-th root. The finite-moment condition makes this value finite between every pair of measures in the class.

Symmetry and nonnegativity follow from the ground metric. A zero value forces identical marginals, using existence or an approximating-coupling argument. To prove the triangle inequality, glue two couplings along their shared intermediate marginal and apply the ground triangle inequality followed by Minkowski’s inequality. Gluing creates a joint law of three positions; it is not the same as assuming the two original couplings were independent. We use the resulting metric theorem rather than reproving the disintegration step here.

In Euclidean space, convergence in Wₚ is equivalent to weak convergence together with convergence of the p-th moments about a fixed reference point. The moment requirement rules out transport cost hidden in an escaping tail. Uniform boundedness of those moments is not sufficient; their actual convergence to the limit moment, or an appropriate uniform-integrability condition, is required. On a common compact support, moments are bounded continuous tests, so weak convergence already supplies the needed condition.

The direct implication can be understood by comparing each measure with a reference atom: Wₚ(μ, δ₀) raised to p equals its p-th moment. The reverse implication uses truncation to a large ball, controls the tail costs by uniform integrability, and applies compact-support transport convergence inside the ball. This proof pattern explains why a numerical domain truncation needs a tail estimate rather than only weak convergence of its renormalized histogram.

A transport problem with cost |x − y|² reports W₂ squared, whereas W₂ itself is its square root. The square is not a metric: among atoms at zero, one, and two, the direct squared cost is four while the sum of the two adjacent squared costs is only two. This does not contradict the metric theorem, whose triangle inequality concerns the rooted quantity. A factor one-half in the cost further rescales the powered value but leaves its optimizing plans unchanged.

Increasing p places greater emphasis on rare long displacements. For fixed measures with enough moments, W₁ is at most W₂ by the usual comparison of L¹ and L² norms under a coupling. Thus W₂ convergence implies W₁ convergence, while the reverse can fail because a sequence retains second-moment tail mass. These comparisons describe the same ground geometry with different integrability demands, not arbitrary scores that can be interchanged without checking assumptions.