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Proof of the Dominated Convergence Theorem 📂Probability Theory

Proof of the Dominated Convergence Theorem

Theorem

Given a probability space $( \Omega , \mathcal{F} , P)$.

If a sequence of random variables $\left\{ X_{n} \right\}_{n \in \mathbb{N}}$ satisfies $n \in \mathbb{N}$ for all $Y \in \mathcal{L}^{1} (\Omega)$ and some $Y \in \mathcal{L}^{1} (\Omega)$, then $$ X_{n} \to X \text{ a.s.} \implies E( X_{n} | \mathcal{G} ) \to \mathcal{G} ) \text{ a.s.} $$


Description

The Dominated Convergence Theorem for conditional expectations, also known as the Conditional Dominated Convergence Theorem (CDCT), tells us that the Dominated Convergence Theorem (DCT) applies to conditional expectations just as it does in other contexts, playing a similar role in probability theory.

Proof

Properties of conditional expectation

  • [7]: $E(X+Y | \mathcal{G}) = E(X | \mathcal{G}) + E(Y| \mathcal{G}) \text{ a.s.}$
  • [10]: $\left| E( X | \mathcal{G} ) \right| \le E ( | X | | \mathcal{G} ) \text{ a.s.}$

$$ \begin{align*} & \left| E( X_{n} | \mathcal{G} ) - E( X | \mathcal{G} ) \right| \\ \color{red}{=}& \left| E( X_{n} - X | \mathcal{G} ) \right| \\ \color{blue}{\le}& E( \left| X_{n} - X \right| | \mathcal{G} ) \\ \le & E \left( \sup_{k \ge n} \left| X_{k} - X \right| | \mathcal{G} \right) \end{align*} $$ So, by the Monotone Convergence Theorem for conditional expectations, properties of lim sup, and condition $X_{n} \to X \text{ a.s.}$, $$ \begin{align*} & \lim_{n \to \infty} \left| E( X_{n} | \mathcal{G} ) - E( X | \mathcal{G} ) \right| \\ \le & \lim_{n \to \infty} E( \sup_{k \ge n} \left| X_{k} - X \right| | \mathcal{G} ) \\ \color{red}{=}& E \left( \lim_{n \to \infty} \sup_{k \ge n} \left| X_{k} - X \right| | \mathcal{G} \right) \\ =& E \left( \lim_{n \to \infty} \left| X_{n} - X \right| \mathcal{G} \right) \\ \color{blue}{=}& 0 \text{ a.s.} \end{align*} $$

See Also