ChapterZero

Recall the Rademacher comparison theorem:

Let be an increasing, convex function, be a compact set, and be real contractions such that , then

where are independent Rademacher (Bernoulli ) variables.

Intuitively, replacing with the identity, this says that if we shrink the coordinates and then take random Bernoulli averages, the expected value of the maximum deviation is going to be less than that for the original coordinates. Therefore, it makes sense to expect that an appropriately modified version of this theorem holds for the case where and the are complex contractions.

In one version of a proof (see “Comparison Theorems, Random Geometry and Some Limit Theorems for Empirical Processes” by Ledoux and Talagrand), an intermediate step is to show that

and their main tool for doing this is a bijection which satisfies, for a given

They prove such a exists without explicitly specifying it, using the marriage theorem.

If I could somehow generalize this to the complex case, i.e. find a bijection which satisfies the above, I’d have at least this intermediate inequality. (As it turns out, this inequality is all I need for my applications, since we only use .) But the question is, is it even reasonable to expect this?

Possibly relevant posts:

• One direction in Khintchine’s inequality for Rademacher sums (9/26/2008)
• Dr. Singh’s Take on the Fourier Unicity Theorem (via Invariant Subspaces) (6/17/2005)
• The other direction in Khintchine’s inequality for Rademacher sums (9/26/2008)