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 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).

The notion of free convolution was introduced by Voiculescu in early 80s in the papers and .

Free Additive Convolution

Let $\mu$ and $\nu$ be two probability measures on the real line, and assume that $X$ is a random variable with law $\mu$ and $Y$ is a random variable with law $\nu$. Assume finally that $X$ and $Y$ are freely independent. Then the free additive convolution$\mu \boxplus \nu$ is the law of $X+Y$.

In many cases, it is possible to compute the probability measure $\mu \boxplus \nu$ explicitly by using complex-analytic techniques and the R-transform of the measures $\mu$ and $\nu$.

Free Multiplicative Convolution

Let $\mu$ and $\nu$ be two probability measures on the the interval $[0,+\infty )$, and assume that $X$ is a random variable with law $\mu$ and $Y$ is a random variable with law $\nu$. Assume finally that $X$ and $Y$ are freely independent. Then the free multiplicative convolution$\mu \boxtimes \nu$ is the law of $X^{1/2}YX^{1/2}$ (or, equivalently, the law of $Y^{1/2}XY^{1/2}$.

A similar definition can be made in the case of laws $\mu ,\nu$ supported on the unit circle $\{z:|z|=1\}$.

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.

As a straightforward example, suppose that A and B are independent large square Hermitian (or symmetric) random matrices, then under some very general conditions, free convolution enables one to:

• Deduce the eigenvalue distribution of A from those of A + B and B.
• Deduce the eigenvalue distribution of A from those of AB and B.

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

• Convolution

• Free probability

References

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

External links

• Alcatel Lucent Chair on Flexible Radio
• http://www.cmapx.polytechnique.fr/~benaych
• http://folk.uio.no/oyvindry