A random matrix model for the Gaussian distribution

Summary

The Gaussian unitary ensemble is a random matrix model (RMM) for the Wigner law. While random matrices in this model are infinitely divisible, the Wigner law is infinitely divisible not in the classical but in the free sense. We prove that any variance mixture of Gaussian distributions -- whether infinitely divisible or not in the classical sense -- admits a RMM of non Gaussian infinitely divisible random matrices. More generally, it is shown that any mixture of the Wigner law admits a RMM. A key role is played by the fact that the Gaussian distribution is the mixture of Wigner law with the <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>2$-gamma distribution.