Probability Theory and Related Fields

, Volume 139, Issue 3, pp 397–413

# The norm of products of free random variables

Article

DOI: 10.1007/s00440-006-0046-x

Kargin, V. Probab. Theory Relat. Fields (2007) 139: 397. doi:10.1007/s00440-006-0046-x

## Abstract

Let Xi denote free identically-distributed random variables. This paper investigates how the norm of products $${\Pi_{n} = X_{1}X_{2}\cdots X_{n}}$$ behaves as n approaches infinity. In addition, for positive Xi it studies the asymptotic behavior of the norm of $${Y_{n} = X_{1}\circ X_{2}\circ \cdots \circ X_{n},}$$ where $${\circ}$$ denotes the symmetric product of two positive operators: $${A\circ B=:A^{1/2}BA^{1/2}}$$ . It is proved that if EXi = 1, then $${\left\Vert Y_{n}\right\Vert }$$ is between $${c_{1}\sqrt{n}}$$ and c2n for certain constant c1 and c2. For $${\left\Vert \Pi_{n}\right\Vert ,}$$ it is proved that the limit of $${n^{-1}\log \left\Vert \Pi _{n}\right\Vert }$$ exists and equals $${\log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }.}$$ Finally, if π is a cyclic representation of the algebra generated by Xi, and if ξ is a cyclic vector, then $${n^{-1}\log \left\Vert \pi \left( \Pi _{n}\right) \xi \right\Vert = \log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }}$$ for all n. These results are significantly different from analogous results for commuting random variables.

46L5415A52