just let them walk together. This is a coupling in the sense that neither particle, taken on its own, can "feel" anything we did. Neither the fact that the other particle follows it 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 in order to obtain that particular result. Biased coins Assume two biased coins, the first with probability p of turning up heads and the second with probability q > p of turning up heads. Intuitively, if both coins are tossed the same number of times, we should expect the first coin turns up fewer heads than the second one. More specifically, for any fixed k, the probability that the first coin produces at least k heads should be less than the probability that the second coin produces at least k heads. However proving such a fact can be difficult with a standard counting argument. Coupling easily circumvents this problem. Let X1, X2, ..., Xn be indicator variables for heads in a sequence of flips of the first coin. For the second coin, define a new sequence Y1, Y2, ..., Yn such that if Xi = 1, then Yi = 1, if Xi = 0, then Yi = 1 with probability (q − p)/(1 − p). Then the sequence of Yi has exactly the probability distribution of tosses made with the second coin. However, because Yi depends on Xi, a toss by toss comparison of the two coins is now possible. That is, for any k ≤ n See also Copula (probability theory) Notes References Probability theory