Probability and Asymptotic Statistics
Y. Eddie Lu, Summer 2026
These notes build the mathematical language behind first-year econometrics: measurability, integration, probability laws, conditioning, convergence, laws of large numbers, weak-limit tools, and central limit theorems.
The aim is not to recognize formulas. It is to know which objects are being compared, which assumption licenses each step, and how to rebuild the proof.
A finished page records available study material, not mastery. Answer every retrieval question before opening its answer. On a second pass, reconstruct the proof blueprints from memory, then open the collapsed proof to check your work.
The nine-lecture route
1. Measure theory
Events, measures, measurable maps, random variables, and probability laws.
2. Integration and measures
Lebesgue integration, limit interchanges, Fubini, and Radon–Nikodym derivatives.
3. Distribution properties
Densities, transformations, moments, inequalities, MGFs, and characteristic functions.
4. Conditional expectation
Information, prediction, independence, iterated expectations, and random assignment.
5. Convergence modes
Almost-sure, probability, \(L^r\), and distributional convergence, plus stochastic orders.
6. CMT and Slutsky
Transform known limits and combine nondegenerate noise with consistent nuisance estimates.
7. Laws of large numbers
When averages estimate population objects, under IID and heterogeneous sampling.
8. Weak-limit tools
Tightness, Portmanteau, Lévy–Cramér, Cramér–Wold, Scheffé, and the delta method.
9. Central limit theorems
Lindeberg arrays, scalar and vector CLTs, studentization, and asymptotic normality.
Dependency map
Objects
Lectures 1–4
Measures, laws, moments, information
\(\longrightarrow\)
Language
Lectures 5–6
Convergence modes and legal transformations
\(\longrightarrow\)
Large samples
Lectures 7–9
Consistency, weak limits, and inference
Three ways to study
Read the motivation, definitions, theorem statements, examples, and common errors. Answer the short retrieval checks, but postpone long proof reconstruction.
State each result from memory. Sketch the proof strategy before opening it, then compare your sketch with the collapsed proof. Reproduce the decisive inequalities or decompositions without copying them.
Use prerequisite retrieval, assumption audits, exit tickets, and mastery checklists. Return to the main text only where retrieval fails.