New York, NY 10033

Probability Theory (Graduate)


Probability Theory (Graduate)

$99.99 $79.99

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.



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