Probabilistic Analysis of Stochastic Processes

Vasan, Dakshesh (2023) Probabilistic Analysis of Stochastic Processes. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS dissertation of Dakshesh Vasan (18MS051)) Thesis_18MS051.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

Gaussian Processes are one of the most important family of stochastic processes given their wide range of use across scientific disciplines and profound implications in Statistics and Mathematics. This report, in three parts, covers some modern developments of tools that are applicable to the analysis of general Gaussian processes. The first part recollects basics, and provides motivation for a ’General Theory’. It also sets the background setup and broad goal of this report - to concern with sample path continuity of general Gaussian processes. The second part introduces a fundamental tool for the modern approach, based on which the two backbone concepts of the general theory - entropy and majorising measures are defined. Both these concepts attempt to measure the ’size’ of a parameter space in different ways. The second part also sets the specific goal for the report - to understand and appreciate a ’Main Entropy Result’ stated immediately after introducing entropy. Following this, essential inequalities of Gaussian process theory are covered and their implications in the analysis of boundedness and suprema distributions are discussed. Boundedness is related to continuity, and this leads to the third part which introduces majorising measures in the context of proving the Main Entropy Result. Other strong characterisations of continuity and boundedness of any general Gaussian process are also noted, following which the report covers specific examples of applications of the Main Entropy Result. The report concludes after having discussed and appreciated several motivations, applications and powerful implications of the entropy-based results while noting the existence of more powerful but harder results based on majorising measures.

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