# Sufficient condition for existence of maximal/optimal coupling

Let us consider two random variables $$X$$ and $$X'$$ defined on the same measurable space $$(E, \xi)$$. Is there a sufficient condition on the distributions of the random variables $$X$$ and $$X'$$ for existence of a maximal coupling $$(\widehat{X}, \widehat{X}')$$ between them?