Measure-theoretic probability

Se basa en

Requisitos previos obligatorios, hasta el fondo: 45

Aprenderás a

• 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.

Definición

• 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.

Teorema

• 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.

Errores frecuentes

• Confusing almost sure convergence with convergence in probability.
• Treating a conditional expectation as a number rather than as a random variable.

Conduce a

Descubrimientos opcionales

• ← Probability

Lecturas complementarias

• David Williams, Probability with Martingales (1991)
• Rick Durrett, Probability: Theory and Examples (1991)
• Patrick Billingsley, Probability and Measure (1979)