Free folding

The free convolution is a binary operation on probability measures on which the addition of free random variables equivalent. ${\displaystyle \mathbb {R} }$

definition

Let and be self-adjoint random variables in a non-commutative probability space , which are free in the sense of free probability theory . Let be the distribution of and the distribution of . Then the distribution of depends only on from and from (and not from the concrete realization of or from ); this distribution of is denoted by and is the free convolution of and . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \mu }$ ${\displaystyle x}$ ${\displaystyle \nu }$ ${\displaystyle y}$ ${\displaystyle x+y}$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x+y}$ ${\displaystyle \mu \boxplus \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$

Free harmonic analysis

The study of the properties of is mostly referred to as free harmonic analysis . There is an advanced theory of the properties of , which often (but not always) runs parallel to the theory of classical convolution . ${\displaystyle \boxplus }$ ${\displaystyle \boxplus }$

literature

• Alexandru Nica, Roland Speicher : Lectures on the Combinatorics of Free Probability (= London Mathematical Society Lecture Note Series. Vol. 335). Cambridge University Press, Cambridge et al. 2006, ISBN 0-521-85852-6 .
• Fumio Hiai, Dénes Petz: The Semicircle Law, Free Random Variables, and Entropy (= Mathematical Surveys and Monographs. Vol. 77). American Mathematical Society, Providence RI 2000, ISBN 0-8218-2081-8 .

Individual evidence

1. ↑ D.-V. Voiculescu, N. Stammeier, M. Weber (eds.): Free Probability and Operator Algebras , Münster Lectures in Mathematics, EMS, 2016, Chapter 6
2. ↑ James A. Mingo, Roland Speicher: Free Probability and Random Matrices . Fields Institute Monographs, Vol. 35, Springer Verlag, New York, 2017, Chapter 3