स्नातकोत्तर प्रथम वर्षProbability & Statisticsपाठ उपलब्ध
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
इस पर आधारित
45 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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.
आम भ्रांतियाँ
- Confusing almost sure convergence with convergence in probability.
- Treating a conditional expectation as a number rather than as a random variable.
आगे ले जाता है
वैकल्पिक खोजें
आगे पढ़ने के लिए
- David Williams, Probability with Martingales (1991)
- Rick Durrett, Probability: Theory and Examples (1991)
- Patrick Billingsley, Probability and Measure (1979)