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Probability and Measure

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Author(s): J. Miller, ed. Dexter Chua
Series: Cambridge Mathematical Tripos Part II Lecture Notes
Publisher: University of Cambridge
Year: 2016

Language: English
City: Cambridge
Tags: maths; mathematics; math; advanced; college; university; higher; further; pure; applied; probability; statistics; stats; stat; measure theory

Introduction
Measures
Measures
Probability measures
Measurable functions and random variables
Measurable functions
Constructing new measures
Random variables
Convergence of measurable functions
Tail events
Integration
Definition and basic properties
Integrals and limits
New measures from old
Integration and differentiation
Product measures and Fubini's theorem
Inequalities and Lp spaces
Four inequalities
Lp spaces
Orthogonal projection in L2
Convergence in L1(P) and uniform integrability
Fourier transform
The Fourier transform
Convolutions
Fourier inversion formula
Fourier transform in L2
Properties of characteristic functions
Gaussian random variables
Ergodic theory
Ergodic theorems
Big theorems
The strong law of large numbers
Central limit theorem
Index