Rate of convergence of random variables for weak convergence

Question

Suppose $X_{n}$ be a sequence of random variable that converges to $X$ in distribution. How can we define the rate of convergence? What would be the reference?

Answer 1

If

$\alpha_n \stackrel{\rightarrow}{_D} \alpha$, s.t.

$\alpha_n \stackrel{_D}{\approx} \alpha + \epsilon(n)$,

then the rate of convergence is the convergence rate of $\epsilon(n)$. Eg, in CLT type convergence, the error would be of order $n^{-\frac{1}{2}}$ and the measures would converge at such rate.