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revised $\mathbb E[\frac{\partial}{\partial\theta}\log f(X;\theta)]^2$ and $\mathbb E[\frac{\partial^2}{\partial\theta^2}\log f(X;\theta)]$
Added note for when $n = 1$, clearly $C(1)$ is not equal to 0.
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comment 3 random numbers to describe point on a sphere
The link here: math.stackexchange.com/questions/44689/… , may be informative for your application.
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