Do all random variables appeared in "convergence of random variable (theory)" share the same probability space? In convergence of random variable (refer to https://en.wikipedia.org/wiki/Convergence_of_random_variables), we're discussing if the sequence of r.v X1, X2, ... ,Xn converges to specific r.v X.
But I'm curious that "X1, X2, ...,Xn and X" are defined over the same probability space.
Or the answer is different according to convergence measure?(e.g a.s convergence, convergence in probability..)
Thank you

It depends on the kind of convergence you're talking about. For  convergence almost surely, the definition doesn't make sense unless the random variables $X_i:\Omega_i \to \mathbb{R}$ live in the same probability space. For convergence in law, you're really talking about the corresponding probability distribution on $\mathbb{R}$, and the definitions make sense even if the $\Omega_i$ differ. Lévy's Continuity Theorem, for example, gives a criterion for convergence in law and allows random variables defined on different probability spaces.
