Infinitesimal Probability Theory with Amalgamation
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Authors
Tseng, Pei-Lun
Date
Type
thesis
Language
eng
Keyword
Operator-Valued Infinitesimal Additive Convolutions , Operator-Valued Infinitesimal Cumulants , Operator-Valued Infinitesimal Central Limit Theorems , Operator-Valued Infinitesimal Free Multiplicative Convolutions
Alternative Title
Abstract
We introduce the notion of operator-valued infinitesimal independence for the free, Boolean, and monotone cases. We show that operator-valued infinitesimal free (respectively Boolean, monotone) independence is equivalent to the operator-valued free (respectively Boolean, monotone) independence over an algebra of $2\times 2$ upper triangular matrices. Then we construct the corresponding operator-valued infinitesimal cumulants for each notion of independence; moreover, we show that the infinitesimal free (respectively Boolean) independence in the operator-valued framework is equivalent to the vanishing of mixed free (respectively Boolean) cumulants and infinitesimal cumulants. In addition, for each notion of independence, we construct the corresponding operator-valued infinitesimal Central Limit Theorem.
We establish formulas to obtain the operator-valued infinitesimal free (respectively Boolean, monotone) additive convolution. Furthermore, for the free case, we also provide the formula for the operator-valued infinitesimal multiplicative convolution.
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Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada
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Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.
ProQuest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.