Probability Theory

Zigang Pan

Chapter Chapter 14 in Measure-Theoretic Calculus in Abstract Spaces, 2023, pp 751-904 from Springer

Abstract: Abstract In this chapter, I study the probability theory. The fundamental notions in probability theory is introduced in Sect. 14.1: random variable, Banach space valued random variable, the expectation, independence of σ-algebras, and independence of random variables, conditional expectation, the law of a random variable, its probability density function, and Fundamental Theorem of Modeling. After a section on Gaussian random variables and vectors, I present the Weak Law of Large Numbers. Then, I directly turn to the study of Martingale theory. I prove the Doob’s Optional Stopping Theorem, Doob’s Upcrossing Lemma, and Doob’s Forward Convergence Theorem. Here, I get another attempt at the Law of Large Numbers, and proved the Strong Law of Large Numbers. I prove Kolmogorov 0-1 Law, some results on the tail σ-algebra, Lévy’s Inversion Formula, the Modes of Convergence Theorem, and Skorokhod Representation Theorem. I then present the Central Limit Theorem, which is dependent on Helly’s Lemma and Lévy’s Convergence Theorem. Then, I prove rigorously the existence of the Wiener process, Wiener process is nowhere differentiable with probability 1, Law of Iterated Logarithms, and results on the stopped Wiener processes. In the last three sections, I study stochastic integration. I present the existence results of Itô process, the Itô’s Formula, and the Girsanov’s Theorem.

Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-21912-2_14

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DOI: 10.1007/978-3-031-21912-2_14

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