🚀 go-pugleaf

My question concerns the application of Multi-resolution stochastic
processes in the area of signal processing.  
Most of the papers say this kind of analysis is
useful for signals where there is no one single characteristic
scale.

o What exactly IS meant by a time series of signals having no
characteristic scale?  
o Does the lack of a unique characteristic scale imply the
non-existence of statistical moments?
o Is there any body of statistical theory existent to detect 
the presence or absence of mutliple scales?
o Is it true that multiplicity of scales imply signals 
with power-law (as opposed to exponential) correlation decay?
 
I know that the issues of long-range range dependence (slow decay of
correlations), 1/f noise, self-similarity, fractal supports, non-analytic
distributions, power laws, wavelets, multi-resolution stochastic
processes etc., are interlinked, but I haven't been able to find any
rigorous (especially in terms of statistical theoretic models)
links.  For eg., Multi-fractal measures are now widely used in
Physics, but I have yet to see them in any journal of statistics.
There have been many journal articles on the application of Chaos
theory to statistical analysis, but all of these concentrate on
invariants like dimension and Lyapunov exponent estimates.  I have
seen NOT ONE article on the consequences on standard statistical tests
and estimation procedures of substituting ,say, fractal spaces for
the standard sample spaces of classical probability analyses. 

Also, can anybody expound on the link of the new theory of Self
Organised Criticality to the presence of multiple scales,if any at
all exist?  Does the SOC model imply that the lack of a
characteristic scale causes a breakdown of the traditional concept
of "Causality", ie., a given input causes a definite response in
output?  This is beginning to sound wrong, but I'd still love to
hear more on this. 

Any leads, clues, explanations, pointers, papers, references, etc.
highly appreciated.  Sorry about the long rather rambling post.

Hari.