Named theorems

My adviser advised me (well, that's his job) to read Asymptotic Statistics by A.W. van der Vaart. Aside from having a really cool name, he does a pretty good job of getting to the point. Lots of results in not much space. As keeping track of named results has served me well over the last year, I will continue to do so. Here's the list from a mere 10 pages (don't really feel like typesetting these, so do your best to wade through the clunky notation):

Portmanteau Lemma

The following are equivalent, describing convergence in distribution:

• P(Xn <= x) -> P(X <= x) for all continuity points of P(X < x).
• E f(Xn) -> E f(X) for all bounded, continuous f
• E f(Xn) -> E f(X) for all bounded Lipschitz functions f
• lim inf E f(Xn) >= E f(X) for all nonnegative, continuous f
• lim inf P(Xn in G) >= P(X in G) for every open set G
• lim sup P(Xn in F) <= P(X in F) for every closed set F
• P(Xn in B) -> P(X in B) fr all Borel sets B with P(X in boundary) = 0

Probably not terribly useful beyond the first three, but who knows?

Continuous Mapping Theorem

Let G be continuous at every point of a set C such that P(X in C) = 1. Then if Xn->X, G(Xn)->G(X). This holds for all three types of convergence (distribution, probability, and almost surely).

I already knew that one, but I don't think I've recorded it here.

Prohorov's Theorem

If Xn converges in distribution to X, then {Xn} is uniformly tight (that is, if there is a uniform N such that for e>0 there exists M such that P(||Xn|| > M) < e.

Conversely, if Xn is uniformly tight, there exists a subsequence that converges in distribution to X.

Not sure how useful that is; maybe it will become obvious in the next 10 pages of the book.

Helly's lemma

Each given sequence Fn of cumulative distribution functions on Rk possesses a subsequence Fnj such that Fnj(x) -> F(x) at each continuity point x. Here, F(x) may be a defective cdf (that is, the density may not integrate to 1).

He also calls out the Markov inequality and Slutsky's lemma, but those two are so well known I won't bother repeating them.

This book has about 450 pages. At this rate, I'll have nearly a thousand results to learn.