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In mathematics, coupling is a proof technique that allows to compare two unrelated variables by "forcing" them to be related in some way. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ...
Definition
Using the standard formalism of probability, let X1 and X2 be two random variables defined on probability spaces (Ω1,F1,P1) and (Ω2,F2,P2). Then a coupling of X1 and X2 is a new probability space (Ω,F,P) over which there are two random variables Y1 and Y2 such that Y1 has the same distribution as X1 while Y2 has the same distribution as X2. Probability theory is the mathematical study of probability. ...
The interesting case is when Y1 and Y2 are not independent.
Examples Assume two particles A and B perform a simple random walk in two dimensions, but 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 A 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. In mathematics and physics, a random walk is a formalization of the intuitive idea of taking successive steps, each in a random direction. ...
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 meet 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 for good. This allows to prove many interesting results that say that "in the long run", it is not important where you started. In mathematics—specifically, in probability theory—the phrase almost surely is a subtle, precise way to say that something is certain except for cases that almost never happen, though still possible. ...
References T. Lindvall, Lectures on the coupling method. Wiley, New York, 1992.
H. Thorisson, Coupling, Stationarity, and Regeneration. Springer, New York, 2000.
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