Theorem (Doob decomposition): Let $(X_n,\mathcal{F}_n)_{n \in \mathbb{N}}$ be a submartingale integrable. Then the Doob decomposition of $X_n$ is given by $$X_n = M_n+A_n$$ where $(M_n)_{n \geq 0}$ a martingale, and $(A_n)_{n \geq 0}$ be an increasing process such that $A_{0}=0$ is $\mathcal{F}_0$-measurable. $A_{n+1}$ is $\mathcal{F}_n$-measurable. and even more : $$ sup_{n} \mathbb {E}|M_n|<+\infty ~and ~ A_{\infty}\in L^1 \Leftrightarrow sup_{n} \mathbb {E}X_n^+<+\infty$$
Show that :
If $(X_n,\mathcal{F}_n)_{n \in \mathbb{N}} $ converge in $L^1 $ if and only if :
$\exists M\in L^1 $ such as: $\forall n\in\mathbb {N}~M_n= \mathbb {E}^{\mathcal{F}_n }M$ and $A_\infty\in L^1$
My effort :
$(\Rightarrow ) $ $(X_n,\mathcal{F}_n)_{n \in \mathbb{N}} $ converge in $L^1 $ then, according to the definition of the limit, we have : $$ sup_{n} \mathbb {E}|X_n|<+\infty $$ According to what is framed above we have: $$ sup_{n} \mathbb {E}|M_n|<+\infty ~~and~~A_{\infty}\in L^1 $$
Now, $(M_n)_n $ is a martingale, then $M_n\to M_{\infty}$ almoust sure and $ M_{\infty}\in L^{1} $.
Let $n\in \mathbb {N}$, $(M_n)_n $ is a martingale, then $\forall p>n~:~~M_n=\mathbb {E}^{\mathcal{F}_n }M_p $ by passage to the limit, we have : $$ M_n=\mathbb {E}^{\mathcal{F}_n }\lim_p M_p =\mathbb {E}^{\mathcal{F}_n }M_{\infty} $$
$(\Leftarrow ) $ we have : $X_n^+=(M_n+A_n)^+\leq M_n^+ +A_n\leq M_n^+ +A_{\infty} $ because $A_n \geq 0$ a.s. Then : $$ \mathbb {E}X_n^+\leq \mathbb {E}M_n+\mathbb {E}A_{\infty}=\mathbb {E}M+\mathbb {E}A_{\infty} $$ Because $M_n= \mathbb {E}^{\mathcal{F}_n }M$ . Then : $$ sup_n \mathbb {E}X_n^+\leq \infty ~~(*) $$
According to $(X_n)_n $ is sub-martingale and $(*) $, we have : $(X_n)_n\to X_\infty $ with $X_\infty\in L^1$ .
My problem is to show that :
$$ \mathbb {E}|X_n- X_\infty|\to 0 $$ And thank you in advance.
I have another idea but I can't get applied, this idea is to show that: $\{X_n~:~n\in \mathbb {N}\}$ is uniformly integrable. We have $\{M_n~:~n \in\mathbb {N}\}$ is uniformly integrable Because $M_n= \mathbb {E}^{\mathcal{F}_n }M$ .
My problem is to show that : $\{A_n~:~n \in\mathbb {N}\}$ is uniformly integrable