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New York, NY 10033
Yiqiao.Yin@YinsCapital.com

Probability Theory (Graduate)

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Probability Theory (Graduate)

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Prerequisites: A thorough knowledge of elementary real analysis and some previous knowledge of probability. Overview of measure and integration theory. Probability spaces and measures, random variables and distribution functions. Independence, Borel-Cantelli lemma, zero-one laws. Expectation, uniform integrability, sums of independent random variables, stopping times, Wald’s equations, elementary renewal theorems. Laws of large numbers. Characteristic functions. Central limit problem; Lindeberg-Feller theorem, infinitely divisible and stable distributions. Cramer’s theorem, introduction to large deviations. Law of the iterated logarithm, Brownian motion, heat equation.

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Description

This is graduate level probability theory notes documented by Yiqiao Yin at Columbia University.

Graduate topics in Probability Theory includes:
– Measure Theory
– Probability Spaces
– Random Variables
– Integration, Properties of the Integral
– Expected Value
– Fubini’s Theory, Laplace Method
– Law of Large Numbers
– Weak LLN
– Borel-Cantelli
– Strong LLN
– Convergence
– Central Limit Theorem
– De Moivre-Laplace Theorem
– Characteristic Functions
– CLT, IID Sequences, Triangular Arrays
– Local Limit Theorems
– Poisson Convergence
– Random Walk
– Stopping Times
– Recurrence
– Martingales
– Polya’s Urn Scheme
– Radon-Nikodym Derivatives
– Branching Processes
– Martingales and Markov Chain

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