Coupling (probability)

In probability theory, coupling is a proof technique that allows one to compare two unrelated variables by "forcing" them to be related in some way.

Definition

Using the standard formalism of probability, let $X_1$ and $X_2$ be two random variables defined on probability spaces $(\Omega_1,F_1,P_1)$ and $(\Omega_2,F_2,P_2)$ . Then a coupling of $X_1$ and $X_2$ is a new probability space $(\Omega,F,P)$ over which there are two random variables $Y_1$ and $Y_2$ such that $Y_1$ has the same distribution as $X_1$ while $Y_2$ has the same distribution as $X_2$ .

The interesting case is when $Y_1$ and $Y_2$ are not independent.

Examples

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 he 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 for 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 just let them walk together.

This is a coupling in the sense that neither particle, taken on its own, can feel anything we did. Nor that fact that the other particle follows him in one way or the other, nor the fact that we changed the coupling rule or when we did it. Each particle performs a simple random walk. And yet, our coupling rule forces them to meet almost surely and to continue from that point on together permanently. This allows one to prove many interesting results that say that "in the long run", it is not important where you started.

References

T. Lindvall, Lectures on the coupling method. Wiley, New York, 1992.
H. Thorisson, Coupling, Stationarity, and Regeneration. Springer, New York, 2000.