In the stats class I took, all the results I have encountered about the convergence in distribution of some random variables are in one way or another consequences of the Central Limit Theorem.
Out of pure curiosity, I wonder whether there exists useful and commonly used results about the convergence in distribution of some statistic $f(X_1,\dots,X_n)$ (where $(X_1,\dots,X_n)$ could be a random sample)
- Which do not follow from the Central limit theorem.
- Which holds for any distribution of the individual $X_i$, or under some "mild" conditions on their distribution.
(sorry for the "mild" I know it means nothing precise, so my appreciation of answers will be partly subjective, but I do not know of a way to formalize this. As an example, the CLT requires that $E(||X_i||^2) < \infty$, that is mild enough for me :).)
I found some interesting examples at http://www.math.uah.edu/stat/dist/Convergence.html, but appart from the CLT, they all violate condition 2 above.
Remark : I realize that "which do not follow from the Central limit theorem" might be a little vague. I do not know how to express this more precisely, but I guess I am curious about results which would not take the form :
- $ X_n $ converges in probability to $X$.
- $Y_n$ converges in distribution to $N(a,V)$ by the CLT.
- So $X_nY_n$ converges in probability to $D \sim X N(a,V)$.
(e.g. Wald statistic converges in distribution to $N(0,A)'A^{-1}N(0,A) \sim \chi^2_q$)
Hope the question is clear enough, feel free to ask for clarifications.