In the mathematical theory of
free probability, the notion of free independence was introduced by
Dan Voiculescu.
[D. Voiculescu, K. Dykema, A. Nica, "Free Random Variables", CIRM Monograph Series, AMS, Providence, RI, 1992] The definition of free independence is parallel to the classical definition of
independence, except that the role of Cartesian products of
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
s (corresponding to
tensor products of their function algebras) is played by the notion of a
free product of (non-commutative) probability spaces.
In the context of Voiculescu's free probability theory, many classical-probability theorems or phenomena have free probability analogs: the same theorem or phenomenon holds (perhaps with slight modifications) if the classical notion of independence is replaced by free independence. Examples of this include: the free central limit theorem; notions of
free convolution Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convoluti ...
; existence of
free stochastic calculus and so on.
Let
be a
non-commutative probability space, i.e. a
unital algebra over
equipped with a
unital linear functional . As an example, one could take, for a probability measure
,
:
Another example may be
, the algebra of
matrices with the functional given by the normalized trace
. Even more generally,
could be a
von Neumann algebra and
a state on
. A final example is the
group algebra of a (discrete)
group with the functional
given by the group trace
.
Let
be a family of unital subalgebras of
.
Definition. The family
is called ''freely independent'' if
whenever
,
and
.
If
,
is a family of elements of
(these can be thought of as random variables in
), they are called
''freely independent'' if the algebras
generated by
and
are freely independent.
Examples of free independence
* Let
be the
free product of groups
, let
be the group algebra,
be the group trace, and set
. Then
are freely independent.
* Let
be
unitary
random matrices, taken independently at random from the
unitary group (with respect to the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
). Then
become asymptotically freely independent as
. (Asymptotic freeness means that the definition of freeness holds in the limit as
).
* More generally, independent
random matrices tend to be asymptotically freely independent, under certain conditions.
References
{{Reflist
Sources
*James A. Mingo, Roland Speicher:
/www.springer.com/us/book/9781493969418 Free Probability and Random Matrices Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
Functional analysis
Free probability theory
Free algebraic structures