{"id":{"repo_id":"queens","oai_identifier":"oai:queensu.scholaris.ca:1974/29059"},"canonical_url":"https://search.dev.ndltd.org/etd/queens/oai:queensu.scholaris.ca:1974/29059","repository":{"repo_id":"queens","name":"Queens University","base_url":"https://qspace.library.queensu.ca/server/oai/request"},"display":{"title":"Infinitesimal Probability Theory with Amalgamation","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.","abstract_html":"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.","abstract_has_math":true,"creators":["Tseng, Pei-Lun"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics and Statistics","school":null,"contributors":[],"advisors":["Mingo, James A","Belinschi, Serban"],"committee_chairs":[],"committee_members":[],"year":null,"date_issued":"","date_published":null,"updated_at":"2026-07-27T20:35:33Z","subjects":["Operator-Valued Infinitesimal Additive Convolutions","Operator-Valued Infinitesimal Cumulants","Operator-Valued Infinitesimal Central Limit Theorems","Operator-Valued Infinitesimal Free Multiplicative Convolutions"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1974/29059","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics and Statistics"]},{"key":"dc:contributor.supervisor","label":"Supervisor","values":["Mingo, James A","Belinschi, Serban"]},{"key":"dc:creator","label":"Author","values":["Tseng, Pei-Lun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2021-08-26T20:50:00Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2021-08-26T20:50:00Z"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Operator-Valued Infinitesimal Additive Convolutions","Operator-Valued Infinitesimal Cumulants","Operator-Valued Infinitesimal Central Limit Theorems","Operator-Valued Infinitesimal Free Multiplicative Convolutions"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1974/29059"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["PhD"]},{"key":"dc:title","label":"Title","values":["Infinitesimal Probability Theory with Amalgamation"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics and Statistics"],"dc:contributor.supervisor":["Mingo, James A","Belinschi, Serban"],"dc:creator":["Tseng, Pei-Lun"],"dc:date.accessioned":["2021-08-26T20:50:00Z"],"dc:date.available":["2021-08-26T20:50:00Z"],"dc:description.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."],"dc:description.degree":["PhD"],"dc:identifier.uri":["http://hdl.handle.net/1974/29059"],"dc:language.iso":["eng"],"dc:subject":["Operator-Valued Infinitesimal Additive Convolutions","Operator-Valued Infinitesimal Cumulants","Operator-Valued Infinitesimal Central Limit Theorems","Operator-Valued Infinitesimal Free Multiplicative Convolutions"],"dc:title":["Infinitesimal Probability Theory with Amalgamation"],"dc:type":["thesis"]},"updated_at":"2026-07-27T20:35:33Z"}