An Invitation to Statistics in Wasserstein Space by Victor M. Panaretos & Yoav Zemel

Author:Victor M. Panaretos & Yoav Zemel
Language: eng
Format: epub, pdf
ISBN: 9783030384388
Publisher: Springer International Publishing

Proof

Assume that F is finite at γ. If θ is any other measure in , write

Since x ≤ 1 + x 2 for all x, the triangle inequality in yields

Since F(γ) < ∞, this shows that F is finite everywhere and the right-hand side vanishes as θ → γ in . Now that we know that F is continuous, the same upper bound shows that it is in fact locally Lipschitz.

Example: let (a n) be a sequence of positive numbers that sum up to one. Let x n = 1∕a n and suppose that Λ equals with probability a n. Then

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