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Wikipedia / 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 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 of random matrices.

The notion of free convolution was introduced by Dan-Virgil Voiculescu.

Free additive convolution

is a random variable in a non commutative probability space are freely independent. Then the free additive convolution. Random matrices 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.

explicitly by using complex-analytic techniques and the R-transform.

Relationship with classical convolution

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Rectangular free additive convolution

has also been defined in the non commutative probability framework by Benaych-Georges and admits the following random matrices 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, then the empirical singular values distribution.

.

Free multiplicative convolution

are freely independent. Then the free multiplicative convolution. Random matrices 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.

supported on the unit circle, with an orthogonal or unitary 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.

  • 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 operators on free groups (Kesten measures); and asymptotic distribution of eigenvalues of sums or products of independent random matrices.

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

The applications in wireless communications, finance and biology have provided a useful framework when the number of observations is of the same order as the dimensions of the system.

See also

References

  • "Free Deconvolution for Signal Processing Applications", O. Ryan and M. Debbah, ISIT 2007, pp.1846–1850

  • James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.

  • D.-V. Voiculescu, N. Stammeier, M. Weber (eds.): Free Probability and Operator Algebras, Münster Lectures in Mathematics, EMS, 2016