Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 348 / 5 / a

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- Solution a

Solution

For

1 \leq p < \infty

, define the probability measures with finite

p

th absolute moment by

P_{p} (X) = {μ \in P (X) : \int_{X} ∣ x - x_{0} ∣^{p} d μ (x) < \infty},

where

x_{0} \in X

is any fixed reference point. The condition is independent of the choice of

x_{0}

. The p-Wasserstein distance is

d_{W_{p}} (μ, ν) = W_{p} (μ, ν) = (π \in Π (μ, ν) in f \int_{X \times X} ∣ x - y ∣^{p} d π (x, y))^{1/ p} .

The infimum is over all transport plans, rather than only transport maps. The product measure gives a finite upper bound using

∣ x - y ∣^{p} \leq 2^{p - 1} (∣ x - x_{0} ∣^{p} + ∣ y - x_{0} ∣^{p}) .

For

p = 1

this is the Wasserstein distance with the metric of Euclidean space; for

p = 2

it is the second Wasserstein distance. “Bounded moment” here means a finite integral, and does not require

X

to be bounded.

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