Probabilistic Operator Algebra Seminar: An introduction to monotonic independence

Seminar | October 2 | 3:45-5:45 p.m. | 748 Evans Hall

 Ian Charlesworth, NSF Postdoctoral Fellow UC Berkeley

 Department of Mathematics

One may think of an "independence relation" as a prescription for building joint distributions of (non-commutative) random variables, satisfying some nice universality properties. Work of Muraki and of Ben Ghorbal and Schurmann has shown that there are very few such universal independences; even with the fewest required "nice properties", there are no more than five. In this talk I will give an introduction to monotonic independence of random variables which fits in only the broadest category as it is not symmetric: $X$ being monotonically independent from $Y$ is \emph {not} equivalent to $Y$ being monotonically independent from $X$. Our goal will be to investigate the behaviour of additive monotonic convolution: given probability measures µ and ν, and random variables $X \sim \mu $ and $Y \sim \nu $ in some algebra where $X$ is monotonically indepenedent from $Y$, what is the distribution of $X+Y$? I will cover the analytic techniques necessary to answer this question, and in the time remaining, begin an investigation into monotone infinite divisibility and semigroups of convolution. This talk will draw material variously from papers of Muraki, of Bercovici and of Hasebe.