Probability Theory and Related Fields

, Volume 139, Issue 3–4, pp 397–413

# The norm of products of free random variables

Article

## 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.

46L54 15A52

## Preview

### References

1. 1.
Bellman R. (1954). Limit theorems for non-commutative operations I. Duke Math. J. 21: 491–500
2. 2.
Bercovici H. and Voiculescu D. (1995). Superconvergence to the central limit and failure of the Cramer theorem for free random variables. Prob. Theory Related Fields 102: 215–222
3. 3.
Cohen J.E. and Newman C.M. (1984). The stability of large random matrices and their products. Ann. Prob. 12: 283–310
4. 4.
Furstenberg H. (1963). Noncommuting random products. Trans. Am. Math. Soc. 108: 377–428
5. 5.
Furstenberg H. and Kesten H. (1960). Products of random matrices. Ann. Math. Statist. 31: 457–469 Google Scholar
6. 6.
Haagerup, U.: On Voiculescus R- and S-transforms for free non-commuting random variables. In: Voiculescu D.-V. (ed.) Free Probability Theory, vol. 12 of Fields Institute Communications, pp. 127–148. American Mathematical Society (1997)Google Scholar
7. 7.
Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables And Entropy, 1st edn., vol. 77 of Mathematical Surveys and Monographs. American Mathematical Society (2000)Google Scholar
8. 8.
Kingman J. F.C. (1973). Subadditive ergodic theory. Ann. Prob. 1: 883–899 Google Scholar
9. 9.
Markushevich A.I. (1979). Theory of Functions of a Complex Variable, 2nd edn. Chelsea Publishing Company, New York Google Scholar
10. 10.
Oseledec V.I. (1968). A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19: 197–231 Google Scholar
11. 11.
Ruelle D. (1982). Characterisic exponents and invariant manifolds in Hilbert space. Ann. Math. 115: 243–290
12. 12.
Ruelle D. (1984). Characterisic exponents for a viscous fluid subjected to time dependent forces. Commun. Math. Phys. 93: 285–300
13. 13.
Voiculescu D. (1987). Multiplication of certain non-commuting random variables. J. Operator Theory 18: 223–235
14. 14.
Voiculescu D. (1991). Limit laws for random matrices and free products. Invent. Math. 104: 201–220
15. 15.
Whittaker E.T. and Watson G.N. (1927). A Course of Modern Analysis, 4 edn. Cambridge University Press, Cambridge Google Scholar