In probability theory, coupling is a proof technique that allows one to compare two unrelated random variables (distributions) and by creating a random vector whose marginal distributions correspond to and respectively. The choice of is generally not unique, and the whole idea of "coupling" is about making such a choice so that and can be related in a particularly desirable way. Definition Using the standard formalism of probability, let and be two random variables defined on probability spaces and . Then a coupling of and is a new probability space over which there are two random variables and such that has the same distribution as while has the same distribution as . An interesting case is when and are not independent. Examples Random walk Assume two particles A and B perform a simple random walk in two dimensions, but they start from different points. The simplest way to couple them is simply to force them to walk together. On every step, if A walks up, so does B, if A moves to the left, so does B, etc. Thus, the difference between the two particles stays fixed. As far as A is concerned, it is doing a perfect random walk, while B is the copycat. B holds the opposite view, i.e. that it is, in effect, the original and that A is the copy. And in a sense they both are right. In other words, any mathematical theorem, or result that holds for a regular random walk, will also hold for both A and B. Consider now a more elaborate example. Assume that A starts from the point (0,0) and B from (10,10). First couple them so that they walk together in the vertical direction, i.e. if A goes up, so does B, etc., but are mirror images in the horizontal direction i.e. if A goes left, B goes right and vice versa. We continue this coupling until A and B have the same horizontal coordinate, or in other words are on the vertical line (5,y). If they never meet, we continue this process forever (the probability of that is zero, though). After this event, we change the coupling rule. We let them walk together in the horizontal direction, but in a mirror image rule in the vertical direction. We continue this rule until they meet in the vertical direction too (if they do), and from that point on, we