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. 



Why do Littlewood-Paley projections behave like iid random variables
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. 


