A free analogue of Hincin's characterization

of infinite divisibility

Authors:
Hari Bercovici and Vittorino Pata

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1011-1015

MSC (1991):
Primary 46L50, 60E07; Secondary 60E10

DOI:
https://doi.org/10.1090/S0002-9939-99-05087-X

Published electronically:
July 28, 1999

MathSciNet review:
1636930

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Hincin characterized the class of infinitely divisible distributions on the line as the class of all distributional limits of sums of infinitesimal independent random variables. We show that an analogue of this characterization is true in the addition theory of free random variables introduced by Voiculescu.

**1.**L. Ahlfors,*Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition.*, McGraw-Hill Book Co., New York, 1953. MR**14:857a****2.**H. Bercovici and V. Pata,*Stable laws and domain of attraction in free probability theory*, with an appendix by Ph. Biane, Ann. of Math. (to appear).**3.**H. Bercovici and D. Voiculescu,*Free convolutions of measures with unbounded support*, Indiana Univ. Math. J.**42**(1993), 733-773. MR**95c:46109****4.**B. V. Gnedenko and A. N. Kolmogorov,*Limit distributions for sums of independent random variables*, Addison-Wesley Publishing Company, Cambridge, Massachusetts, 1954. MR**16:52d****5.**H. Maassen,*Addition of freely independent random variables*, J. Funct. Anal.**106**(1992), 409-438. MR**94g:46069****6.**V. Pata,*Domains of partial attraction in noncommutative probability*, Pacific J. Math.**176**(1996), 235-248. MR**98g:46100****7.**D. Voiculescu,*Symmetries of some reduced free product C-algebras*, Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics, No. 1132, Springer Verlag, New York, 1985, pp. 556-588. MR**87d:46075****8.**D. Voiculescu,*Addition of certain non-commuting random variables*, J. Funct. Anal.**66**(1986), 323-346. MR**87j:46122**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46L50,
60E07,
60E10

Retrieve articles in all journals with MSC (1991): 46L50, 60E07, 60E10

Additional Information

**Hari Bercovici**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Email:
bercovic@indiana.edu

**Vittorino Pata**

Affiliation:
Dipartimento di Matematica, Università di Brescia, Brescia 25123, Italy

Email:
pata@ing.unibs.it

DOI:
https://doi.org/10.1090/S0002-9939-99-05087-X

Received by editor(s):
May 13, 1998

Published electronically:
July 28, 1999

Additional Notes:
The first author was partially supported by a grant from the National Science Foundation.

Communicated by:
Dale Alspach

Article copyright:
© Copyright 2000
American Mathematical Society