研究生一年级Probability & Statistics计划中
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
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前置知识
45 项必需的前置知识（一直追溯到底）
你将学会
- 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.
定义
- 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.
定理
- 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.
常见误解
- Confusing almost sure convergence with convergence in probability.
- Treating a conditional expectation as a number rather than as a random variable.
通向
可选发现
延伸阅读
- David Williams, Probability with Martingales (1991)
- Rick Durrett, Probability: Theory and Examples
- Patrick Billingsley, Probability and Measure (1979)