# Free convolution

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'''Free convolution''' is the [free probability](/source/free_probability) analog of the classical notion of [convolution](/source/convolution) of [probability measures](/source/Probability_measure). Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free multiplicative convolution can be reduced to additive convolution by passing to logarithms of random variables). These operations have some interpretations in terms of [empirical spectral measures](/source/empirical_spectral_measures) of [random matrices](/source/random_matrices).<ref>Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. {{isbn|978-0-521-19452-5}}.</ref>

The notion of free convolution was introduced by [Dan-Virgil Voiculescu](/source/Dan-Virgil_Voiculescu).<ref>Voiculescu, D., Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), 323–346</ref><ref>Voiculescu, D., Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), 2223–2235</ref>

== Free additive convolution ==

Let <math>\mu</math> and <math>\nu</math> be two probability measures on the real line, and assume that <math>X</math> is a [random variable](/source/random_variable) in a non commutative [probability space](/source/probability_space) with law <math>\mu</math> and <math>Y</math> is a random variable in the same non commutative probability space with law <math>\nu</math>.  Assume finally that <math>X</math> and <math>Y</math> are [freely independent](/source/free_independence).  Then the '''free additive convolution''' <math>\mu\boxplus\nu</math> is the law of <math>X+Y</math>. [Random matrices](/source/Random_matrices) interpretation: if <math>A</math> and <math>B</math> are some independent <math>n</math> by <math>n</math> Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the [empirical spectral measures](/source/empirical_spectral_measures) of <math>A</math> and <math>B</math> tend respectively to <math>\mu</math> and <math>\nu</math> as <math>n</math> tends to infinity, then the empirical spectral measure of <math>A+B</math> tends to <math>\mu\boxplus\nu</math>.<ref>Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. {{isbn|978-0-521-19452-5}}.</ref>

In many cases, it is possible to compute the probability measure <math>\mu\boxplus\nu</math> explicitly by using complex-analytic techniques and the [R-transform](/source/R-Transform) of the measures <math>\mu</math> and <math>\nu</math>.

== Relationship with classical convolution ==
There exists a precise functional relationship between the free convolution and classical convolution of compactly-supported probability measures: the expectation of <math>f</math> over the free convolution <math>\mu\boxplus\nu</math>  is at most the expectation of <math>f</math> over the classical convolution, provided that the fourth derivative of <math>f</math> is non-negative. The non-negativity of the fourth derivative is also a necessary condition for the comparison to hold true for all compactly-supported <math>\mu</math> and <math>\nu</math>.<ref>{{cite arXiv |last=Heinävaara |first=Otte |title=Convolution comparison measures |date=2026-02-10 |class=math.FA |eprint=2602.10373}}</ref>

== Rectangular free additive convolution ==

The rectangular free additive convolution (with ratio <math>c</math>) <math>\boxplus_c</math> has also been defined in the non commutative probability framework by Benaych-Georges<ref>Benaych-Georges, F., Rectangular random matrices, related convolution, Probab. Theory Related Fields Vol. 144, no. 3 (2009) 471-515.</ref> and admits the following [random matrices](/source/random_matrices) interpretation. For <math>c\in [0,1]</math>, for  <math>A</math> and <math>B</math> are some independent <math>n</math> by <math>p</math> complex (resp. real) random matrices such that at least one of them is invariant, in law, under multiplication on the left and on the right by any unitary (resp. orthogonal) matrix and such that the [empirical singular values distribution](/source/empirical_singular_values_distribution) of <math>A</math> and <math>B</math> tend respectively to <math>\mu</math> and <math>\nu</math> as <math>n</math> and <math>p</math> tend to infinity in such a way that <math>n/p</math> tends to <math>c</math>, then the [empirical singular values distribution](/source/empirical_singular_values_distribution) of <math>A+B</math> tends to <math>\mu\boxplus_c\nu</math>.<ref>Benaych-Georges, F., Rectangular random matrices, related convolution, Probab. Theory Related Fields Vol. 144, no. 3 (2009) 471-515.</ref>

In many cases, it is possible to compute the probability measure <math>\mu\boxplus_c\nu</math> explicitly by using complex-analytic techniques and the rectangular R-transform with ratio <math>c</math> of the measures <math>\mu</math> and <math>\nu</math>.

== Free multiplicative convolution ==

Let <math>\mu</math> and <math>\nu</math> be two probability measures on the interval <math>[0,+\infty)</math>, and assume that <math>X</math> is a random variable in a non commutative probability space with law <math>\mu</math> and <math>Y</math> is a random variable in the same non commutative probability space with law <math>\nu</math>.  Assume finally that <math>X</math> and <math>Y</math> are [freely independent](/source/free_independence).  Then the '''free multiplicative convolution''' <math>\mu\boxtimes\nu</math> is the law of <math>X^{1/2}YX^{1/2}</math> (or, equivalently, the law of <math>Y^{1/2}XY^{1/2}</math>.  [Random matrices](/source/Random_matrices) interpretation: if <math>A</math> and <math>B</math> are some independent <math>n</math> by <math>n</math> non negative Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the [empirical spectral measures](/source/empirical_spectral_measures) of <math>A</math> and <math>B</math> tend respectively to <math>\mu</math> and <math>\nu</math> as <math>n</math> tends to infinity, then the empirical spectral measure of <math>AB</math> tends to <math>\mu\boxtimes\nu</math>.<ref>Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. {{isbn|978-0-521-19452-5}}.</ref>

A similar definition can be made in the case of laws <math>\mu,\nu</math> supported on the [unit circle](/source/unit_circle) <math>\{z:|z|=1\}</math>, with an orthogonal or unitary [random matrices](/source/random_matrices) interpretation.

Explicit computations of multiplicative free convolution can be carried out using complex-analytic techniques and the S-transform.

== Applications of free convolution ==

* Free convolution can be used to give a proof of the free [central limit theorem](/source/central_limit_theorem).
* Free convolution can be used to compute the laws and spectra of sums or products of random variables which are free.  Such examples include: [random walk](/source/random_walk) operators on free groups (Kesten measures); and [asymptotic distribution](/source/asymptotic_distribution) of eigenvalues of sums or products of independent [random matrices](/source/random_matrix).

Through its applications to random matrices, free convolution has some strong connections with other works on G-estimation of Girko.

The applications in [wireless communication](/source/wireless_communication)s, [finance](/source/finance) and [biology](/source/biology) have provided a useful framework  when the number of observations is of the same order as the dimensions  of the system.

== See also ==

* [Convolution](/source/Convolution)
* [Free probability](/source/Free_probability)
* [Random matrix](/source/Random_matrix)

==References==

{{Reflist}}

* "Free Deconvolution for Signal Processing Applications", O. Ryan and M. Debbah, ISIT 2007, pp.&nbsp;1846–1850
*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.
*D.-V. Voiculescu, N. Stammeier, M. Weber (eds.): [http://www.ems-ph.org/books/book.php?proj_nr=208 Free Probability and Operator Algebras], Münster Lectures in Mathematics, EMS, 2016

==External links==
*[http://www.supelec.fr/d2ri/flexibleradio/Welcome.html Alcatel Lucent Chair on Flexible Radio]
*[http://www.cmapx.polytechnique.fr/~benaych http://www.cmapx.polytechnique.fr/~benaych]
*[http://folk.uio.no/oyvindry http://folk.uio.no/oyvindry]
* [https://www.math.uni-sb.de/ag/speicher/speicher_publikationenE.html survey articles] of Roland Speicher on free probability.

Category:Signal processing
Category:Combinatorics
Category:Functional analysis
Category:Free probability theory
Category:Free algebraic structures

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Adapted from the Wikipedia article [Free convolution](https://en.wikipedia.org/wiki/Free_convolution) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Free_convolution?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.