Convergence in distribution results not deriving from central limit theorem?

Question

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)

1. Which do not follow from the Central limit theorem.
2. 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.

Answer 1

I used to ask myself the same question and I recently had an answer in an Extreme Value Theory course.

In this domain we focus on extreme values i.e. on $max_i(X_i)$ not the sum. (from wikipedia)

The Fisher–Tippett–Gnedenko theorem states: If there exist sequences of constants $a_n>0$ and $b_n\in \mathbb R$ such that $P((M_n-b_n)/a_n \leq z) \rightarrow G(z) \propto \exp (-(1+\zeta z)^{-1/\zeta} ) $ as $n \rightarrow \infty$ where $\zeta$ depends on the tail shape of the distribution. When normalized, ''G'' belongs to one of the following non-degenerate distribution families:

Weibull, law: $ G(z) = \begin{cases} \exp\left\{-\left( -\left( \frac{z-b}{a} \right) \right)^\alpha\right\} & z<b \\ 1 & z\geq b \end{cases}$

When the distribution of $M_n$ has a light tail with finite upper bound (or finite support). Also known as Type 3.

Gumbel,law: $G(z) = \exp\left\{-\exp\left(-\left(\frac{z-b}{a}\right)\right)\right\}\text{ for }z\in\mathbb R.$

When the distribution of $M_n$ has an exponential tail. Also known as Type 1

Frechet, law: $G(z) = \begin{cases} 0 & z\leq b \\ \exp\left\{-\left(\frac{z-b}{a}\right)^{-\alpha}\right\} & z>b. \end{cases}$ when the distribution of $M_n$ has a heavy tail (including polynomial decay). Also known as Type 2.

In all cases, $\alpha>0$.

In practice you have to find $a_n$ et $b_n$ then $\zeta$ to find the type of your initial distribution.

Some interesting results from my course:

Distrib.F: law of Max, law of Min.

Gaussian: Gumbel, Gumbel

Cauchy: Frechet, Frechet

Uniform: Weibull, Weibull

Pareto: Frechet, Weibull

Gamma: Gumbel, Weibull

Exponential: Gumbel, Weibull

If Y Weibull, log(Y) Gumbel and $Y^{-1}$ Fréchet.

The lognormal and the χ2 distributions belong to the Gumbel domain (seems both min and max)

Answer 2

I am not sure there is an answer to this question in the affirmative.

If you don't know the distribution of the individual X's then how can you make any statement about the distribution of some statistic of which they are a function?

Also, I don't think it is correct to say that the distribution of some some statistic follows from the Central Limit Theorem. The convergence properties arise because of the topology of the space you're working in, i.e. a sigma algebra on a Borel set. That's where the convergence properties come from, and it is because of the convergence properties that the Central Limit Theorem is valid.

Perhaps I am not understanding your question, but if you are looking for distribution-free (or nearly distribution free results), then a non-parametric textbook may hold the answer to your question.