Determinantal point processes

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dc.contributor.advisorOrtega Cerdà, Joaquim
dc.contributor.authorJiménez Lumbreras, Rubén
dc.date.accessioned2026-03-19T15:31:30Z
dc.date.available2026-03-19T15:31:30Z
dc.date.issued2025-09-28
dc.descriptionTreballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2026. Director: Joaquim Ortega Cerdà
dc.description.sponsorshipIn this Master’s Final Project, we develop a survey of determinantal point processes, which form a family of simple point processes whose characteristic property is that their joint intensities $\rho_k(x_1,\ldots,x_k)=\det(K(x_i,x_j))_{1\le i,j\le k}\ \ \forall k\ge1\text{ and }x_1,\ldots,x_k\in E$ are given by the determinants of a kernel $K$ in the appropriate space $E$. Intuitively, these processes describe and allow us to model the behaviour of objects or points that verify certain repulsion interactions between them, determined by their respective kernels. After introducing the necessary preliminaries and using them to define this type of processes, then we proceed to review some of their most remarkable properties. Among them, we especially focus on their existence and uniqueness results, as well as on other selected properties that describe probabilistic behaviours of these processes and justify their interest. Finally, we delve into some specific examples of determinantal point processes, with the aim of showing that they appear naturally in various fields, such as in the study of certain random matrices, whose eigenvalues are examples and give rise to processes of the family. In addition, examples of these processes have also recently appeared in the development of machine learning solutions, something that we also explore in the last section of the project.
dc.format.extent84 p.
dc.format.mimetypeapplication/pdf
dc.identifier.urihttps://hdl.handle.net/2445/228328
dc.language.isoeng
dc.rightscc by-nc-nd (c) Rubén Jiménez Lumbreras, 2026
dc.rights.accessRightsinfo:eu-repo/semantics/openAccess
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/
dc.sourceMàster Oficial - Matemàtica Avançada
dc.subject.classificationProbabilitats
dc.subject.classificationProcessos estocàstics
dc.subject.classificationEstadística matemàtica
dc.subject.classificationIntel·ligència artificial
dc.subject.classificationTreballs de fi de màster
dc.subject.classificationRubén Jiménez Lumbreras
dc.subject.otherProbabilities
dc.subject.otherStochastic processes
dc.subject.otherMathematical statistics
dc.subject.otherArtificial intelligence
dc.subject.otherMaster's thesis
dc.titleDeterminantal point processes
dc.typeinfo:eu-repo/semantics/masterThesis

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