# optimal coupling

Given probability distributions $(\mu_1, \ldots, \mu_n)$ on a nice state space $E$ is it always possible to find a random vector $(X_1, \ldots, X_n)$ such that $(X_k, X_{k+1})$ is an optimal coupling of $\mu_k$ and $\mu_{k+1}$ for any $1 \leq k \leq n-1$? For example, this is true for Gaussian distributions $\mu_k \sim \mathcal{N}(\alpha_k, \sigma_k^2)$.

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The gluing Lemma.

Let $(X_i , μ_i)$, $i = 1, 2, 3$ be Polish probability spaces. If $(X_1,X_2 )$ is a coupling of $(μ_1 , μ_2 )$ and $(Y_2 , Y_3 )$ is a coupling of $(μ_2,μ_3 )$, then one can construct a triple of random variables $(Z_1 , Z_2 , Z_3 )$ such that $(Z_1 , Z_2 )$ has the same law as $(X_1 , X_2 )$ and $(Z_2 , Z_3 )$ has the same law as $(Y_2 , Y_3 )$.

Therefore, it suffices to have an optimal coupling between each pair $(\mu_k,\mu_{k+1})$.

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Thank you -- it is exactly what I was looking for. –  Alekk Mar 17 '11 at 22:31
You're welcome. This can be found, for example, in the book by Cédric Villani "Optimal transport, old and new". –  camomille Mar 17 '11 at 22:36