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Coupling (probability)

In probability theory, coupling is a proof technique that allows one to compare two unrelated random variables(distributions) X {displaystyle X} and Y {displaystyle Y} by creating a random vector W {displaystyle W} whose marginal distributions correspond to X {displaystyle X} and Y {displaystyle Y} respectively. The choice of W {displaystyle W} is generally not unique, and the whole idea of 'coupling' is about making such a choice so that X {displaystyle X} and Y {displaystyle Y} can be related in a particularly desirable way.Using the standard formalism of probability, let X 1 {displaystyle X_{1}}   and X 2 {displaystyle X_{2}}   be two random variables defined on probability spaces ( Ω 1 , F 1 , P 1 ) {displaystyle (Omega _{1},F_{1},P_{1})}   and ( Ω 2 , F 2 , P 2 ) {displaystyle (Omega _{2},F_{2},P_{2})}  . Then a coupling of X 1 {displaystyle X_{1}}   and X 2 {displaystyle X_{2}}   is a new probability space ( Ω , F , P ) {displaystyle (Omega ,F,P)}   over which there are two random variables Y 1 {displaystyle Y_{1}}   and Y 2 {displaystyle Y_{2}}   such that Y 1 {displaystyle Y_{1}}   has the same distribution as X 1 {displaystyle X_{1}}   while Y 2 {displaystyle Y_{2}}   has the same distribution as X 2 {displaystyle X_{2}}  .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.Copula (probability theory)

[ "Algebra of random variables", "Regular conditional probability", "Probability mass function", "Probability measure", "Random variable" ]
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