Coupling (probability)

In probability theory, coupling is a proof technique that allows one to compare two unrelated random variables (distributions) X and Y by creating a random vector W whose marginal distributions correspond to X and Y respectively. The choice of W is generally not unique, and the whole idea of "coupling" is about making such a choice so that X and Y can be related in a particularly desirable way. 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. 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.

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Related publications (1)

Limit Behaviour of the Voter Model, Exclusion Process and Regenerative Chains

Samuel Schöpfer

Though the following topics seem unlinked, most of the tools used in this thesis are related to random walks and renewal theory. After introducing the voter model, we consider the parabolic Anderson model with the voter model as catalyst. In GÄRTNER, DEN HOLLANDER and MAILLARD [44], the behaviour of the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant with respect to the catalyst, was investigated. It was shown that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant. In Chapter 3 we address some questions left open in this paper by considering specifically when the Lyapunov exponents are the a priori maximal value. Then, we use exclusion process techniques to show that the evolution of a perturbed threshold voter model is recurrent in the critical case. The key to our approach is to develop the ideas of BRAMSON and MOUNTFORD [9] : we exhibit a Lyapunov-Foster function for the discrete time version of the process. We also make a widespread use of coupling arguments. Finally, using the regenerative scheme of COMETS, FERNÁNDEZ and FERRARI [19], we establish a functional central limit theorem for discrete time stochastic processes with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the limit theorem is fulfilled.

EPFL2012