The Exponential Map in Non-commutative Probability | Zendy

Michael Anshelevich | Zendy; Octavio Arizmendi | Zendy

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The Exponential Map in Non-commutative Probability

Author(s) -

Michael Anshelevich,

Octavio Arizmendi

Publication year - 2016

Publication title -

international mathematics research notices

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.757

H-Index - 76

eISSN - 1687-0247

pISSN - 1073-7928

DOI - 10.1093/imrn/rnw164

Subject(s) - mathematics , multiplicative function , semigroup , commutative property , convolution (computer science) , homomorphism , monotone polygon , probability measure , discrete mathematics , exponential function , mathematical proof , class (philosophy) , transformation (genetics) , pure mathematics , combinatorics , mathematical analysis , biochemistry , chemistry , geometry , machine learning , artificial intelligence , artificial neural network , computer science , gene

The wrapping transformation $W$ is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class $\mathcal{L}$ of measures, $W$ also transforms the three non-commutative convolutions---free, Boolean, and monotone---to their multiplicative counterparts. Moreover, the restriction of $W$ to $\mathcal{L}$ preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of numerous known, and new, results about multiplicative convolutions.

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