TITLE: Why do Littlewood-Paley projections behave like iid random variables
QUESTION [13 upvotes]: 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.

REPLY [5 votes]: 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
http://www.math.unm.edu/~crisp/papers/princeton.pdf
or my own lecture notes at
http://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.