Let be the identity matrix with components.

We want to find a coupling of and where . For all , denote the density of . The reflection coupling is based on the following result: Set as the orthogonal reflection wrt to .

Since , we get . Then, is equivalent to . Moreover, so that . Then, Now, noting that is an isometry and that , we get which implies that and . Finally, which concludes the proof.

We now intend to construct a coupling of and . We use the Lemma with and where to construct a coupling and we set and . This is is equivalent to the following coupling. Write the density of ,

• Draw independently and
• Set and set
  • if
  • otherwise.