Why do Littlewood-Paley projections behave like iid random variables

Question

I have read more than once that the Littlewood-Paley (LP) projections of a function (i.e. decomposing a function into parts with frequency localization in different octaves) behave in some sense like iid random variables.

I am also aware of some facts (like inequalities for square functions vs. Khinchine Inequality) which "look similar".

Is there any precise way of stating this similarity?

And why do we have this similarity?

Can we somehow interpret the LP projections as something like independent random variables?

A related question concerns systems of functions of the form

$$ f_k(\cdot):=f(n_k \cdot )\quad {k\geq 1} , $$

with $(n_k)_{k\geq 1}$ a lacunary sequence. Also in this case (under suitable assumptions) the functions $f_k$ behave like iid random variables in the sense that they satisfy the Central Limit Theorem and the LIL.

Answer 1

It is much better to replace 'iid random variables' above by 'martingale differences.'

The usual Littlewood-Paley square function is closely related to the Haar square function.
And the Haar square function is exactly a martingale square function, namely a sum of squares of martingale differences.

One can pass back and forth, from martingale to continuous analogs. A striking method to do this was found by Stefanie Petermichl, when she found a simple way to obtain the Hilbert transform from a modification of a martingale multiplier.

Answer 2

If one replaces the real line with the Walsh ring $F_2[t](\frac{1}{t})$ (or equivalently, replaces the Fourier transform by the Fourier-Walsh transform), then Littlewood-Paley projections become precisely the same thing as martingale differences. See for instance the lecture notes of Pererya and Ward at

https://web.archive.org/web/20210507042436/http://www.math.unm.edu/~crisp/papers/princeton.pdf

or my own lecture notes at

https://www.math.ucla.edu/~tao/254a.1.01w/notes5.ps

Very roughly speaking, the difference between the two is the difference between a sine wave and a square wave - and the latter, when viewed in binary, depicts the fluctuation of a random bit. In contrast, a sine wave of frequency comparable to $2^k$ (and more generally, a Littlewood-Paley projection to that range of frequencies) depends primarily, but not exclusively, of the $k^{th}$ bit in the binary expansion of the domain variable - and one can view the binary bits of the domain variable as independent random variables.

Answer 3

There is a quantitative way to express the somewhat vague notion of "almost independence of the Littlewood-Paley projections".

Let $\mathcal F_n$, $n\in\mathbb Z$, be the minimal $\sigma$-algebra generated by the set $\mathcal D_n$ of dyadic cubes in $\mathbb R^d$ $$\mathcal D_n=\left\{\prod\limits_{k=1}^{d}[m_k2^{-n},(m_k+1)2^{-n})|\quad (m_1,\dots,m_d)\in\mathbb Z^d\right\}.$$ Then for any locally integrable function $f$ on $\mathbb R^d$, one may define the conditional expectation $E_n(f)$ with respect to the filtration of $\sigma$-algebras $\{\mathcal F_k|\ k\in\mathbb Z \}$: $$E_n(f)=\sum\limits_{Q\in \mathcal D_n}\chi_Q\ \frac{1}{|Q|}\int_Q f(x)dx.$$ It is not hard to check that the differences $D_n(f)=E_n(f)-E_{n-1}(f)$, $n\in\mathbb Z$, define a martingale. This means that the family of Haar functions has the martingale property (and they indeed can be viewed as iid random variables).

Now, the Littlewood-Paley projections $\Delta_n$ (and partial sums of Fourier series, in general) cannot be interpreted directly as conditional expectations. However, they do behave almost like the family of Haar functions. Roughly speaking, the families of projections $\{\Delta_k\}_{k\in\mathbb Z}$ and $\{D_j\}_{j\in\mathbb Z}$ are almost biorthogonal.

Theorem. There exists a constant $C$ such that for every $k$, $j\in\mathbb Z$ the following estimate on the operator norm of $D_k\Delta_j:\ L^2(\mathbb R^n)\to L^2(\mathbb R^n)$ is valid $$\|D_k\Delta_j\|=\|\Delta_jD_k\|\leq C2^{-|j-k|}.$$

This result is relatively recent and is due to Grafakos and Kalton (see Chapter 5 of the book by Grafakos).

Answer 4

I am not completely sure of the connection. Eli Stein mentioned it briefly in his class one time, which is why I know of a reference, but I cannot expand on the answer. A good starting point is to look at Rademacher Functions. There is a nice way to prove the Littlewood-Paley Square Function estimate using Rademacher functions (see, e.g. Stein, Singular Integrals and Differentiability Properties of Functions, section 5.2). The Rademacher functions, on the other hand, finds use in probability theory, see, e.g. http://www.statslab.cam.ac.uk/~james/Lectures/pm6.pdf ). This is where, however, my really limited knowledge on this subject ends.

Edit: You may also be interested in Stein's Topics in Harmonic Analysis related to Littlewood-Paley Theory. The theory is based upon diffusive semigroups, but has some connections (IIRC) in the spirit of Doob to martingales and ergodic theorems. I don't have my copy handy at the moment, though.