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Lecture 3-3. Sigma Algebra(σ-algebra)

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1. Sigma Algebra

2. Random Variable

3. Filtration

1. Sigma Algebra

⑴ Probability Space (Ω, ℱ)

① Ω: Sample Space

② ℱ: Sigma Algebra (σ-algebra, event space), i.e., a collection of subsets of Ω

○ Example 1. When Ω = {1, 2, 3}, the σ-algebra ℱ = {∅, Ω} corresponds to the case of knowing nothing

○ Example 2. When Ω = {1, 2, 3}, the σ-algebra ℱ = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, Ω} corresponds to the case of being able to see all events

○ Example 3. When Ω = {1, 2, 3}, the σ-algebra ℱ = {∅, {1}, {2, 3}, Ω} represents an intermediate case

③ ω ∈ Ω: Realized sample. In a random process, it means a sample path.

⑵ Algebra

① Condition 1. non-empty: Ω ∈ ℱ or ∅ ∈ ℱ holds

② Condition 2. Closed under complement: If A ∈ ℱ, then A^C = Ω - A ∈ ℱ also holds

○ Considering together with the non-empty condition implies that ∅ and Ω must be elements of ℱ

③ Condition 3. Closed under finite union: If A, B ∈ ℱ, then A ∪ B ∈ ℱ also holds

○ If A₁, ⋯, A_n ∈ ℱ, then ∪_i A_i = A₁ + ⋯ + A_n ∈ ℱ also holds

⑶ Sigma Algebra (σ-algebra)

① Motivation: When the sample space is very large (e.g., Ω = ℝ), formal probability theory does not apply well (e.g., ℱ = 2^Ω), so it is necessary to restrict ℱ to a sigma algebra. Related to Caratheodory’s extension theorem.

② Condition 1. Must be an algebra

③ Condition 2. Closed under countably infinite unions: For A_i ∈ ℱ, ∪_i A_i = A₁ + ⋯ + A_∞ ∈ ℱ

④ Borel σ-algebra: The smallest sigma algebra containing all open sets

○ Ω = ℝ, ℱ = ℬ(ℝ)

○ Using the properties of sigma algebra, open intervals → closed intervals, half-open intervals, singletons {x}, [1,3] ∪ [4,5] are also included in the Borel algebra

○ Complicated sets like the set of rationals and irrationals are also Borel sets

○ More complex sets obtained by countably many unions, differences, or intersections of intervals are all Borel sets

○ Not limited only to ℝ; can be defined for any topological space X: for example, on [0,1], on ℝⁿ, or on any general topological space, each has its own Borel σ-algebra

○ In fact, there exist sets such as Lebesgue non-measurable sets that cannot be made by the Borel σ-algebra: related to uncountable infinity

⑤ Intuitive meaning of sigma algebra

○ A collection of subsets on a non-empty set Ω

○ The set of all events to which probability can be assigned

○ The set of all functions/random variables that can be generated

2. Random Variable

⑴ Probability Distribution

① A function that assigns values to elements of ℱ, i.e., ℙ: ℱ ↦ [0, 1]

② Condition 1. ℙ(Ω) = 1

③ Condition 2. For countably infinite, mutually exclusive {A_i}_i∈ℕ, ℙ(A₁ + ⋯ + A_∞) = ℙ(A₁) + ⋯ + ℙ(A_∞)

○ Disjoint: A_i ∩ A_j = ∅

⑵ Random Variable (measurable function)

① Expression 1. If there exists a function X such that ^∀B ∈ ℬ(ℝ), X^-1(B) ∈ ℱ (measurable), then that function is called a random variable

② Expression 2. X: Ω → ℝ is a random variable ⇔ X^-1(A) = {ω ∈ Ω: X(ω) ∈ A} ∈ ℱ ∀ A ∈ ℬ(ℝ)

○ Meaning 1. Existence of inverse image: i.e., ℬ(ℝ) is ℱ-measurable. If X is continuous, this usually holds.

○ Meaning 2. Existence of range of inverse image: i.e., the inverse image’s range is a subset of ℱ

○ ℱ can be viewed as equivalent to the collection of all random variables (or functions) measurable with respect to it

○ Example: ℙ_X(x ∈ A) = ℙ(X^-1(A))

③ General measurable space

○ Definition: If X: Ω → Ω₁ between two measurable spaces (Ω, ℱ) and (Ω₁, ℱ₁) satisfies the following condition, then X is called a random variable

④ Random Process (stochastic process)

○ Definition: X: ℐ × Ω ↦ E, where for each i ∈ ℐ, there exists a random variable X(i, ·): Ω ↦ E

⑶ π-class and λ-class

① Definition of π-class: If A, B ∈ 𝒞 ⊂ 2^Ω, then A ∩ B ∈ 𝒞

② Definition of λ-class

③ Property of λ-class

④ Dynkin’s theorem: If 𝒟 is a π-class, 𝒞 is a λ-class, and 𝒟 ⊂ 𝒞, then σ(𝒟) ⊂ 𝒞

⑷ Stationary

① Strictly stationary

② Wide-sense stationary: Strictly stationary is also wide-sense stationary

⑸ Independence

① Definition of independence using joint distribution: ℙ(X₁ ∈ B₁, X₂ ∈ B₂) = ℙ(X₁ ∈ B₁) ℙ(X₂ ∈ B₂) ∀B₁, B₂ ∈ ℬ(ℝ)

② Definition of independence using moments

③ Definition of independence using moment generating function

④ Definition of independence using σ-algebra: σ(x₁) and σ(x₂) are independent (where σ(X) = {X^-1(A): A ∈ ℬ(ℝ)})

⑹ Markov Process

① Bayes’ Rule: ℙ(A | B) = ℙ(A ∩ B) / ℙ(B) if ℙ(B) > 0

② Conditional expectation 𝔼[X | 𝒢]

③ Markov process: ∀A ∈ 𝓔, ℙ(X_in ∈ A | X_i1, X_i2, ⋯, X_in-1) = ℙ(X_in ∈ A | X_in-1), i.e., the current state depends only on the immediately preceding state

3. Filtration

⑴ Doob’s Theorem

① σ(X₁, X₂, ···, X_n): The smallest σ-algebra making X₁, X₂, ···, X_n measurable

② Doob’s Theorem: σ(X₁, X₂, ···, X_n) is equivalent to the collection of all functions of the form g(X₁, X₂, ···, X_n)

③ The larger the σ-algebra, the greater the number of measurable functions with respect to it — i.e., more information

⑵ Filtration

① A collection of σ-algebras arranged in an increasing order by inclusion

② Ordered by ⊆, and if ℱ₁ ⊆ ℱ₂, then ℱ₂ is after ℱ₁

③ For convenience, with time index t = 0, 1, 2, ⋯, filtration is {ℱ_t}_t∈ℤ⁺, satisfying ℱ_s ⊆ ℱ_t for all s ≤ t

④ Intuitive meaning: Represents a situation where information increases as time passes and observations accumulate

⑶ Martingale

① Property of conditional expectation

○ For any random variable Y, 𝔼[Y | X₁, ···, X_n] = 𝔼[Y | σ(X₁, ···, X_n)] holds

○ Reason: σ(X₁, ···, X_n) is equivalent to the set of all functions generated by X₁, ···, X_n

② Martingale: A stochastic process {X_t}_t∈ℤ⁺ adapted to filtration {ℱ_t}_t∈ℤ⁺ satisfies all the following conditions

○ Condition 1. For all t ∈ ℤ⁺, X_t is ℱ_t-measurable

○ Condition 2. For all t ∈ ℤ⁺, 𝔼[|X_t|] is finite

○ Condition 3. For all t ∈ ℤ⁺, 𝔼[X_t | ℱ_s] = X_s, almost surely for all s ≤ t

③ Note: An i.i.d. process is generally not a martingale (except when it is a constant process)

Input: 2025.09.07 08:40