Suppose that $X, X^\prime$ are two random variables with the same distribution $\mathcal{N}(\mu, \sigma^2)$.
Is there anything that we can say about the "distance" between the two random variables?
e.g., $\|X - X^\prime\|$
The thing is that I don't know if the space of random variables is a metric or normed space (never talked about in class), and the types of norms that are typically placed on these functions.
More generally, what does it mean for a function of a random variable to be a Lipschitz function?
Let $f$ be a function of $X$, then when I say that it is Lipschitz, does it mean $$\|f(X) - f(X^\prime)\| \leq L \|X - X^\prime\|$$ or $$\|f(x) - f(x^\prime)\| \leq L \|x - x^\prime\|,$$ where the lower case are the realizations.