MU-404 Measure-Theoretic Probability

Unit · year 4

MU-404 · Measure-Theoretic Probability

Threads chance25 lectures5 theorems

Probability rebuilt on measure theory, up to the central limit theorem and martingales.

PREREQUISITES

Lectures

L01	Probability Spaces as Measure Spaces —
L02	Random Variables as Measurable Functions —
L03	Distribution Functions and Laws —
L04	Expectation as an Integral —
L05	Independence and Product Measures —
L06	The Borel–Cantelli Lemmas The Borel–Cantelli lemmas
L07	Kolmogorov's Zero-One Law The Borel–Cantelli lemmas
L08	Modes of Convergence —
L09	Almost-Sure versus In-Probability Convergence The Borel–Cantelli lemmas
L10	Kolmogorov's Inequality The strong law of large numbers
L11	The Strong Law of Large Numbers The strong law of large numbers
L12	Applications of the Strong Law The strong law of large numbers
L13	Characteristic Functions The central limit theorem
L14	Lévy's Continuity Theorem The central limit theorem
L15	Weak Convergence and Tightness The central limit theorem
L16	The Central Limit Theorem The central limit theorem
L17	Lindeberg's Condition and Extensions The central limit theorem
L18	Conditional Expectation: Existence Conditional expectation as projection
L19	Conditional Expectation as Projection Conditional expectation as projection
L20	Properties of Conditional Expectation Conditional expectation as projection
L21	Filtrations and Martingales The martingale convergence theorem
L22	Stopping Times and Optional Stopping The martingale convergence theorem
L23	Doob's Inequalities The martingale convergence theorem
L24	The Martingale Convergence Theorem The martingale convergence theorem
L25	Synthesis: Measure Theory as the Grammar of Chance The strong law of large numbers, The central limit theorem

Theorems in this unit

T-131

The Borel–Cantelli lemmas

When infinitely many events occur, almost surely or not.

T-132

The strong law of large numbers

Sample means converge almost surely to the mean.

T-133

The central limit theorem

Sums of independent variables are asymptotically Gaussian.

T-134

Conditional expectation as projection

Conditioning is orthogonal projection in L2.

T-135

The martingale convergence theorem

A bounded martingale converges almost surely.