First-year graduateProbability & StatisticsPlanned
Measure-theoretic probability
Probability on measure-theoretic foundations: probability spaces, independence, modes of convergence, conditional expectation, martingales, and proofs of the strong law of large numbers and the central limit theorem.
math.prob.measure-theoretic-probability
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 45
You will learn to
- Prove the Borel–Cantelli lemmas and the strong law of large numbers.
- Define conditional expectation via the Radon–Nikodym theorem and use its properties.
- Prove the central limit theorem using characteristic functions, and apply the martingale convergence theorem.
Definition
- A probability space is a measure space (Ω, F, P) with P(Ω) = 1.
- For an integrable X and a sub-σ-algebra G, the conditional expectation E[X | G] is the (almost surely unique) G-measurable random variable whose integral over every set in G equals that of X.
- An adapted integrable sequence (Xₙ) is a martingale if E[Xₙ₊₁ | Fₙ] = Xₙ almost surely.
Theorem
- Strong law of large numbers: for independent identically distributed integrable Xᵢ, the sample means converge almost surely to E[X₁].
- Lévy's continuity theorem relates convergence in distribution to convergence of characteristic functions.
- Doob's martingale convergence theorem: an L¹-bounded martingale converges almost surely.
Common misconceptions
- Confusing almost sure convergence with convergence in probability.
- Treating a conditional expectation as a number rather than as a random variable.
Leads to
Optional discoveries
Further reading
- David Williams, Probability with Martingales (1991)
- Rick Durrett, Probability: Theory and Examples
- Patrick Billingsley, Probability and Measure (1979)