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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?

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Again, since you are asking for a definition and a reference I have added both those tags. –  Alex Becker Dec 28 '11 at 22:29
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up vote 0 down vote accepted

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.

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Epsilon is rendered wrong as I must have a latex error...apologies. –  gnometorule Dec 28 '11 at 23:12
    
...as is -1/2 exponent, at least on my phone.... –  gnometorule Dec 28 '11 at 23:14
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