Martingale convergence theorem for $L^2$

Question

Let $(\Omega, F, P)$ be probability space with probability measure $P$.

Theorem

Let $X\in L^1(P)$, let $F_k$ be an increasing family of sigma algebras, $F_k \subset F$ and $F=\cup_{k=1}^{\infty} \sigma(F_k)$. Then, $$E[X|F_k] \to E[X|F] \mbox{ as $k \to \infty$},$$ a.e. $P$ and in $L^1(P)$.

I want to use this theorem for $X\in L^2(P).$ First Since $X\in L^2(P)$, $X\in L^1(P)$.

So, $$E[X|F_k] \to E[X|F] \mbox{ as $k \to \infty$},$$ a.e. $P$ and in $L^1(P)$.

But, I wanna show that $E[X|F_k] \to E[X|F]$ also in $L^2(P)$.

Could you help me?

Answer 1

1. By fact 6 of this answer, it suffices to establish that the set of random variables $\{Y_n,n\geqslant 1\}$ where $$ Y_n=\left(\mathbb E\left[X\mid\mathcal F_n\right]-\mathbb E\left[X\mid\mathcal F\right]\right)^2, $$ is uniformly integrable.
2. Since $\mathbb E\left[X\mid\mathcal F\right]$ is square integrable, it suffices to establish uniform integrability of $\{Y'_n,n\geqslant 1\}$ where $$ Y'_n=\left(\mathbb E\left[X\mid\mathcal F_n\right] \right)^2. $$
3. By Jensen's inequality, it suffices to establish uniform integrability of $\{Y''_n,n\geqslant 1\}$ where $$ Y''_n= \mathbb E\left[X^2\mid\mathcal F_n\right] . $$
4. Boundedness in $L^1$ of $\{Y''_n,n\geqslant 1\}$ is clear; if $A$ is a measurable set, then for all $R$, $$ \mathbb E\left[\mathbb E\left[X^2\mid\mathcal F_n\right]\mathbf 1_A\right]= \mathbb E\left[\mathbb E\left[X^2\mathbf 1\{X^2\leqslant R\}\mid\mathcal F_n\right]\mathbf 1_A\right]+\mathbb E\left[\mathbb E\left[X^2\mathbf 1\{X^2> R\}\mid\mathcal F_n\right]\mathbf 1_A\right]\leqslant R\Pr(A)+\mathbb E\left[ X^2\mathbf 1\{X^2> R\} \right]. $$

Answer 2

By Jensen's inequality

$$(E[X|F_n])^2 \leq E[X^2|F_n]$$

Taking Expectation

$$E(E[X|F_n])^2 \leq E[X^2]$$

Then $\sup _{n\geq 0}E(E[X|F_n])^2 < \infty$

By, Martingale $L^p$ convergence Theorem,

$E[X|F_n] \to E[X|F]$ almost surely and in $L^2$.