From my understanding of coupling in statistics it can be defined as a procedure to devise a valid joint distribution from marginal distributions. However, I can't find further information on this topic nor could I find any worked examples online whether it be a discrete or continuous case.
Can anyone suggest any resources i.e textbooks/online sources so that I can get a better understanding of this topic?
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There is a book: Coupling, stationarity, and regeneration by Hermann Thorisson. Although this text may be quite advanced.
Quick simple example of coupling: take two distinct fair coins, $X$ and $Y$. We will simulate them using a discrete uniform random number generator $U$. When $U=0$, we will say that $X$ is heads and $Y$ is tails, and when $U=1$, will say that $X$ is tails and $Y$ is heads. We have coupled $X$ and $Y$. They retain their individual distributions but have a joint distribution on a common probability space according to $U$.
This post, about a proof that uses coupling, might be helpful. The answers provide commentary on the motivation and the logic behind the coupling. (But note there's an error at the end of the proof, which is discussed here.)