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The Dependence Structure of Negatively Dependence Measures

Barzegar, Milad | 2022

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 56648 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Alishahi, Kasra; Zamani, Mohammad Sadegh
  7. Abstract:
  8. Strongly Rayleigh measures are an important class of negatively dependent (repulsive) probability measures. These measures are defined via a geometric condition, called “real stability”, on their generating polynomials, and have interesting probabilistic properties. On one important property of negatively dependent measures is their rigid dependence structure. In other words, the it is impossible for these measures to have strong overall dependencies. In this thesis, we study two manifestations if this phenomenon: (1) paving property and (2) tail triviality. Informally, the paving property states that it is possible to partition the set of the components of every strongly Rayleigh random vector to a small number of subsets such that the elements of each set are “almost independent”. An important step in the proof of this result is generalizing the paving conjecture to real stable polynomials. The paving conjecture is one of the equivalents of the Kadison-Singer problem. This problem, which remained open for 54 years, was resolved in 2013 by Marcus, Spielman and Srivastava. In the second part of this thesis, we prove that the tail sigma-field of every sequence of negatively associated Bernoulli random variables that satisfies the “summable covariances property” is trivial---that is, every tail event has probability zero or one. A corollary of this result is the tail triviality of strongly Rayleigh stochastic processes. We also study the tail properties of negatively associated Gaussian and Gaussian threshold processes. We prove that these stochastic processes have trivial tail sigma-fields, although they do not necessarily satisfy the summable covariances property. Using this fact, we construct negatively associated Gaussian threshold measures that are not strongly Rayleigh. Thus we answer a question of Pemantle which asks for “natural classes” of negatively associated measures that are not strongly Rayleigh
  9. Keywords:
  10. Negative Dependence ; Strongly Rayleigh Measures ; Kadison-Singer Problem ; Paving Conjecture ; Gaussian Threshold Processes ; Zero-One Theorems ; Tail Sigma-Field

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