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 $\mu\_k \sim \mathcal{N}(\alpha\_k, \sigma\_k^2)$.



optimal coupling
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 $(\mu\_k,\mu\_{k+1})$.


