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Proof of Lévy's Theorem in Probability Theory 📂Probability Theory

Proof of Lévy's Theorem in Probability Theory

Theorem

Let’s assume that a probability space (Ω,F,P)( \Omega , \mathcal{F} , P) is given.

If η\eta is an integrable random variable and {Fn}n∈N\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}} is a sequence of sigma fields where {Fn}n∈N\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}} is Fn⊂Fn+1\mathcal{F}_{n} \subset \mathcal{F}_{n+1}, then n→∞n \to \infty when E(η∣Fn)→E(η∣F∞) E \left( \eta | \mathcal{F}_{n} \right) \to E \left( \eta | \mathcal{F}_{\infty} \right)

  • F∞=⨂n=1∞Fn\displaystyle \mathcal{F}_{\infty} = \bigotimes_{n=1}^{\infty} \mathcal{F}_{n} does not mean the tensor product but represents the smallest sigma field containing all elements of Fn\mathcal{F}_{n}. It’s not particularly new, as the smallest sigma field containing all open sets of a topological space Ω\Omega has been referred to as the Borel sigma field. However, if it’s difficult, it could simply be taken as a sigma field satisfying the filtration condition.

Explanation

Unlike the Lebesgue’s theorem in measure theory, where the integrand remains static and the sigma field expands, the essence is not significantly different.

Lévy’s 0-1 Law

Lévy’s theorem, also known as Lévy’s zero–one law, suggests that the conditional probability P(A∣F∞)=E(1A∣F∞)P \left( A | \mathcal{F}_{\infty} \right) = E \left( 1_{A} | \mathcal{F}_{\infty} \right) of an event A∈F∞A \in \mathcal{F}_{\infty} is almost surely P(A∣Fn)→1A∈{0,1} P \left( A | \mathcal{F}_{n} \right) \to 1_{A} \in \left\{ 0 , 1 \right\} That is, either 00 or 11 when n→∞n \to \infty. Intuitively, the fact that the sigma field expands while satisfying Fn⊂Fn+1\mathcal{F}_{n} \subset \mathcal{F}_{n+1}, resulting in the filtration of sigma fields, means an increase in the amount of information, which clarifies whether the event AA occurs as either 00 or 111.

Proof

Strategy: It’s necessary to employ the properties of regular martingales in probability theory and Doob’s martingale convergence theorem. Note that since η\eta is only given as an integrable random variable, it cannot assert that it’s almost surely X∞=ηX_{\infty} = \eta for all A∈F∞A \in \mathcal{F}_{\infty} as it’s not F∞\mathcal{F}_{\infty}-measurable. However, according to the properties of conditional expectation, E(η∣F∞)E \left( \eta | \mathcal{F}_{\infty} \right) is F∞\mathcal{F}_{\infty}-measurable regardless of what η\eta is, and the equation to be proved actually becomes X∞=E(η∣Fn)X_{\infty} = E \left( \eta | \mathcal{F}_{n} \right).

Claim: Let’s define Xn:=E(η∣Fn)X_{n} : = E \left( \eta | \mathcal{F}_{n} \right) and a random variable X∞:=lim⁡n→∞Xn\displaystyle X_{\infty} := \lim_{n \to \infty} X_{n} that is F∞\mathcal{F}_{\infty}-measurable. It’s necessary to prove X∞=E(η∣F∞)X_{\infty} = E \left( \eta | \mathcal{F}_{\infty} \right).

Part 1. E(X∞∣Fn)=XnE(X_{\infty} | \mathcal{F}_{n}) = X_{n}

According to the definition of XnX_{n}, {(Xn,Fn)}\left\{ (X_{n} , \mathcal{F}_{n} ) \right\} is a regular martingale. Therefore, it’s a uniformly integrable martingale, and becomes a L1\mathcal{L}_{1} convergent martingale, converging to X∞X_{\infty} in the sense of L1\mathcal{L}_{1}. Furthermore, since {(Xn,Fn)}\left\{ (X_{n} , \mathcal{F}_{n} ) \right\} is a closable martingale, E(X∞∣Fn)=XnE(X_{\infty} | \mathcal{F}_{n}) = X_{n} is obtained.

Now, to use Doob’s martingale convergence theorem, the following definitions are introduced.

Pi system and Lambda system:

  1. A P\mathcal{P} satisfying the following is called a π\pi-system. A,B∈P  ⟹  A∩B∈P A, B \in \mathcal{P} \implies A \cap B \in \mathcal{P}
  2. A L\mathcal{L} satisfying the following conditions is called a λ\lambda-system.
  • (i): ∅∈L\emptyset \in \mathcal{L}
  • (ii) A∈L  ⟹  Ac∈LA \in \mathcal{L} \implies A^{c} \in \mathcal{L}
  • (iii) For all i≠ji \ne j, when Ai∩Aj=∅\displaystyle A_{i} \cap A_{j} = \emptyset {An}n∈N⊂L  ⟹  ⋃n∈NAn∈L\displaystyle \left\{ A_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{L} \implies \bigcup_{n \in \mathbb{N}} A_{n} \in \mathcal{L}

Part 2. L:={A∈F:∫AX∞dP=∫AηdP}\displaystyle \mathcal{L} := \left\{ A \in \mathcal{F} : \int_{A} X_{\infty} dP = \int_{A} \eta dP \right\} is a Lambda system

  • Part 2-(i). ∅∈L\emptyset \in \mathcal{L}

  • Part 2-(ii). A∈L  ⟹  Ac∈LA \in \mathcal{L} \implies A^{c} \in \mathcal{L}

    • As Part 1 implied E(X∞∣Fn)=XnE(X_{\infty} | \mathcal{F}_{n}) = X_{n}, according to the definition of conditional expectation, ∫ΩX∞dP=∫ΩE(X∞∣Fn)dP=∫ΩXndP=∫ΩE(η∣Fn)dP=∫ΩηdP \begin{align*} \int_{\Omega} X_{\infty} dP =& \int_{\Omega} E ( X_{\infty} | \mathcal{F}_{n} ) dP \\ =& \int_{\Omega} X_{n} dP \\ =& \int_{\Omega} E \left( \eta | \mathcal{F}_{n} \right) dP \\ =& \int_{\Omega} \eta dP \end{align*} thus Ω∈L\Omega \in \mathcal{L}, and by the definition of L\mathcal{L}, if A∈LA \in \mathcal{L}, ∫AcX∞dP=∫ΩX∞dP−∫AX∞dP=∫ΩηdP−∫AηdP=∫AcηdP \begin{align*} \int_{A^{c}} X_{\infty} dP =& \int_{\Omega} X_{\infty} dP - \int_{A} X_{\infty} dP \\ =& \int_{\Omega} \eta dP - \int_{A} \eta dP \\ =& \int_{A^{c}} \eta dP \end{align*} This calculation was possible because the probability measure PP excludes cases like ∞−∞\infty - \infty being a finite measure. Therefore, by the calculation Ac∈LA^{c} \in \mathcal{L}, and in conclusion, A∈L  ⟹  Ac∈L A \in \mathcal{L} \implies A^{c} \in \mathcal{L}

  • Part 2-(iii). For all i≠ji \ne j, when Ai∩Aj=∅\displaystyle A_{i} \cap A_{j} = \emptyset {An}n∈N⊂L  ⟹  ⋃n∈NAn∈L\displaystyle \left\{ A_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{L} \implies \bigcup_{n \in \mathbb{N}} A_{n} \in \mathcal{L}

    • ∫⋃i∈NAiX∞dP=∑i=1∞∫AiX∞dP=∑i=1∞∫AiηdP=∫⋃i∈NAiηdP\begin{align*} \int_{\bigcup_{i \in \mathbb{N}} A_{i}} X_{\infty} dP =& \sum_{i=1}^{\infty} \int_{A_{i}} X_{\infty} dP \\ =& \sum_{i=1}^{\infty} \int_{A_{i}} \eta dP \\ =& \int_{\bigcup_{i \in \mathbb{N}} A_{i}} \eta dP \end{align*}

Part 3. P:=⋃n∈NFn\displaystyle \mathcal{P} := \bigcup_{n \in \mathbb{N}} \mathcal{F}_{n} is a Pi system

If A,B∈⋃n∈NFn\displaystyle A, B \in \bigcup_{n \in \mathbb{N}} \mathcal{F}_{n}, then for some n1,n2∈Nn_{1}, n_{2} \in \mathbb{N}, A∈Fn1B∈Fn2 A \in \mathcal{F}_{n_{1}} \\ B \in \mathcal{F}_{n_{2}} Therefore, (A∩B)∈Fmax⁡{n1,n2}⊂⋃n∈NFn (A \cap B) \in \mathcal{F}_{\max \left\{ n_{1} , n_{2} \right\} } \subset \bigcup_{n \in \mathbb{N}} \mathcal{F}_{n}

Part 4. P⊂L\displaystyle \mathcal{P} \subset \mathcal{L}

To say that A∈P=⋃n∈NFn\displaystyle A \in \mathcal{P} = \bigcup_{n \in \mathbb{N}} \mathcal{F}_{n} is to suggest the existence of some n0∈Nn_{0} \in \mathbb{N} that satisfies A∈Fn0A \in \mathcal{F}_{n_{0}}. Hence, since Part 1 showed {(Xn,Fn)}\left\{ (X_{n} , \mathcal{F}_{n} ) \right\} was a martingale, ∫AXmdP=∫AE(Xm∣Fn0)dP=∫AXn0dP=∫AE(η∣Fn0)dP=∫AηdP \begin{align*} \int_{A} X_{m} dP =& \int_{A} E ( X_{m}| \mathcal{F}_{n_{0}} ) dP \\ =& \int_{A} X_{n_{0}} dP \\ =& \int_{A} E \left( \eta | \mathcal{F}_{n_{0}} \right) dP \\ =& \int_{A} \eta dP \end{align*} Therefore, ∣∫AXmdP−∫AX∞dP∣≤∫A∣Xm−X∞∣dP≤E∣Xm−X∞∣ \begin{align*} \left| \int_{A} X_{m} dP - \int_{A} X_{\infty} dP \right| \le & \int_{A} \left| X_{m} - X_{\infty} \right| dP \\ \le & E | X_{m} - X_{\infty} | \end{align*} Here, since {(Xn,Fn)}\left\{ (X_{n} , \mathcal{F}_{n} ) \right\} is a martingale converging in the sense of L1\mathcal{L}_{1}, when m→∞m \to \infty, E∣Xm−X∞∣→0E| X_{m} - X_{\infty}| \to 0 and lim⁡m→∞∫AXmdP=∫AX∞dP \lim_{m \to \infty} \int_{A} X_{m} dP = \int_{A} X_{\infty} dP And for all m∈Nm \in \mathbb{N}, ∫AXmdP=∫AE(η∣Fm)dP=∫AηdP \int_{A} X_{m} dP = \int_{A} E \left( \eta | \mathcal{F}_{m} \right) dP = \int_{A} \eta dP Therefore, ∫AX∞dP=lim⁡m→∞∫AXmdP=lim⁡m→∞∫AηdP=∫AηdP \begin{align*} \int_{A} X_{\infty} dP =& \lim_{m \to \infty} \int_{A} X_{m} dP \\ =& \lim_{m \to \infty} \int_{A} \eta dP \\ =& \int_{A} \eta dP \end{align*} Thus, by the definition of L\mathcal{L}, A∈LA \in \mathcal{L}. In conclusion, A∈P  ⟹  A∈LP⊂L A \in \mathcal{P} \implies A \in \mathcal{L} \\ \mathcal{P} \subset \mathcal{L}

Part 5.

Doob’s martingale convergence theorem: If a Pi system P\mathcal{P} is a subset of a Lambda system L\mathcal{L}, there exists a sigma field σ(P)\sigma ( \mathcal{P} ) satisfying P⊂σ(P)⊂L\mathcal{P} \subset \sigma ( \mathcal{P} ) \subset \mathcal{L}.

σ(P)=σ(⋃n∈NFn)=⨂n=1∞Fn=F∞ \begin{align*} \sigma \left( \mathcal{P} \right) =& \sigma \left( \bigcup_{n \in \mathbb{N}} \mathcal{F}_{n} \right) \\ =& \bigotimes_{n=1}^{\infty} \mathcal{F}_{n} \\ =& \mathcal{F}_{\infty} \end{align*} According to Doob’s martingale convergence theorem, there exists a sigma field F∞\mathcal{F}_{\infty} satisfying the following: A∈F∞  ⟹  ∫AX∞dP=∫AηdP A \in \mathcal{F}_{\infty} \implies \int_{A} X_{\infty} d P = \int_{A} \eta dP Therefore, for all A∈F∞A \in \mathcal{F}_{\infty}, ∫AX∞dP=∫AηdP=∫AE(η∣F∞)dP \int_{A} X_{\infty} d P = \int_{A} \eta dP = \int_{A} E \left( \eta | \mathcal{F}_{\infty} \right) dP Since X∞X_{\infty} and E(η∣F∞)E \left( \eta | \mathcal{F}_{\infty} \right) are F∞\mathcal{F}_{\infty}-measurable, according to the properties of the Lebesgue integral, almost surely X∞=E(η∣F∞)X_{\infty} = E \left( \eta | \mathcal{F}_{\infty} \right). Then, as initially defined, XnX_{n} and X∞X_{\infty}, lim⁡n→∞E(η∣Fn)=lim⁡n→∞Xn=X∞=E(η∣F∞) \begin{align*} \lim_{n \to \infty} E \left( \eta | \mathcal{F}_{n} \right) =& \lim_{n \to \infty} X_{n} \\ =& X_{\infty} \\ =& E \left( \eta | \mathcal{F}_{\infty} \right) \end{align*}

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See Also

  1. https://en.wikipedia.org/wiki/Doob%27s_martingale_convergence_theorems#L%C3%A9vy's_zero%E2%80%93one_law ↩︎