PROBABILISTIC MODELS FOR DEPENDENT VARIABLES: COMPLEX ANALYTIC AND COMBINATORIAL APPROACHES

In this thesis, we study probabilistic models for dependent variables using complex analytic and combinatorial approaches. Our study consists of two independent parts. In the first part, we establish central limit theorems (CLTs) under zero-free conditions, building upon a quantitative extension of Marcinkiewicz's Theorem. In particular, we obtain quantitative decay estimates for the Kolmogorov-Smirnov distance between a real-valued random variable X and a Gaussian under the condition that the characteristic function of X does not vanish only in a bounded disk. This leads to a quantitative CLT applicable to very general and possibly strongly dependent random systems. In addition, we generalize our CLT to multi-dimensional cases and propose a variant of this generalization with more relaxed zero-free conditions. In the second part, we investigate an interesting connection between the concepts of vines in probability theory and MAT-labeled graphs in hyperplane arrangement theory. We show that there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mucksch for the characterization of MAT-freeness in terms of partially ordered sets.

Author dc:creator

• TRAN MANH HUNG

• poset
• labeled graph
• MAT-labeling
• vine
• central limit theorem
• Marcinkiewicz's theorem

Licence

• https://scholarbank.nus.edu.sg/bitstreams/a9e1d20a-1215-4597-9cb3-2ff5d5b8b47b/download