Measure concentration is currently a hot topic
in probability theory (cf. Talagrand 1995, 1996).
A general description of problems
pertinent to that topic would lead too far,
but one result now considered as an early example
of a measure concentration theorem is well known
to information theorists. It is the blowing up lemma
(Margulis 1974, Ahlswede, Gács and Körner 1976)
that says, intuitively, that by slightly ``blowing up''
any set 
of not exponentially
small probability, one gets a set of probability close to 1.
In IT, the blowing up lemma was originally just a mathematical tool, particularly useful in multiuser Shannon theory (cf. Csiszár and Körner 1981). Today it is an integral part of IT due to its information theoretic proof (Marton 1986) that was, actually, its first simple proof. Recently, Marton extended her approach to prove also other measure concentration results, providing beautiful examples of applications of IT in probability theory. Here we sketch some main ideas of that approach, restricting attention to the simplest case.
Let 
be PD's on a finite set 
, let
, and let
denote the 
-blow up of a set
(where d denotes normalized Hamming distance).
Marton (1996) proved that the distance
of two subsets of 
can be bounded in terms of their
probabilities as
With the choice 
, when
, (2.15) gives the following
strong version of the blowing up lemma:
Sketch of proof of (2.15):
(i) The key idea is to show that given and any PD 
on
not necessarily of product form, there exist random variables
with distribution 
and
with distribution such that
For n=1, this is obvious by Pinsker's inequality (1.4),
since the minimum of 
subject to 
,
equals 
.
For n>1, the proof of (2.17) goes by induction,
using a coupling argument to extend 
and 
satisfying the induction hypothesis by suitable
new components 
.
(ii) The result (i) implies, by the triangle inequality,
that given arbitrary PD's and 
on
, there exist and 
with distributions and such that
(iii) Finally, (2.15) follows from (2.18), letting
and
be the conditional distributions obtained from
by conditioning on A and B, respectively.
Indeed,
with that choice, the left hand side of (2.18)
is lower bounded
by d(A,B) since 
with
probability 1.
This approach to derive bounds like (2.15) has been extended
to distributions of certain processes with memory in the role
of , including mixing Markov chains (Marton 1996).
So far, other approaches to measure concentration could not
be so extended. It should be noted that a weaker asymptotic
form of the blowing up lemma has been established for a
rather broad class of processes, again with substantial use
of IT ideas (Marton and Shields 1994).
