Coupling of several random variables

Weconsiderthe problem offinding conditions under which an $\alpha$-coupling is possible for several random variables $X_1, X_2,\dots, X_k$ with a finite or countably infinite range of values and with given probability distributions, i.e., the possibility of constructing a joint distribution of these random variables such that $\mathrm{Pr}\{X_1=X_2=\dots= X_k\} = \alpha$.