For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which converges to 0 "singular". I have also seen them called "potentials".
Is there a good reference for these
two facts?
Do these decompositions have standard
names?
Is there a standard term for a martingale which converges to 0?
Below, $\Vert M \Vert$ is the $L^1$-bound of the martingale $M_k$.
Decomposition 1. Let $(M_{k})$ be an $L^{1}$-bounded martingale with respect to the filtration $({\mathcal{F}}_{k})$. Then there are two nonnegative martingales $(P_k)$ and $(N_k)$ such that such that $M_{k}=P_k-N_k$ a.e. for all $k$, and $\left\Vert M\right\Vert =\left\Vert P\right\Vert +\left\Vert N\right\Vert = \Vert P_0 \Vert_1 + \Vert N_0 \Vert_1$.
Further, this decomposition is a.e. unique;
$(P_k)=\sup_{n\geq k}E[[M_{n}]^{+}\mid\mathcal{F}_{k}]$ a.e.;
$N_k=\sup_{n \geq k}E[[M_{n}]^{-}\mid\mathcal{F}_{k}]$ a.e.;
$\lim_{k\rightarrow\infty}P_k=[\lim_{k}M_{k}]^{+}$ a.e.; and
$\lim_{k\rightarrow\infty}N_k=[\lim_{k}M_{k}]^{-} a.e.$
Decomposition 2. Let $(M_{k})$ be an $L^{1}$-bounded martingale with respect to the
filtration $(\mathcal{F}_{k})$ and let $M_{\infty}=\lim_{n}M_{n}$.
Then there is a uniformly integrable martingale $(U_k)$ and
a singular martingale $(S_k)$ such that $M_{k}=U_k+S_k$
a.e. for all $k$.
Further, this decomposition is a.e. unique;
$U_k=E[M_{\infty}\mid\mathcal{F}_{k}]$ a.e.;
$S_k=E[M_{k}-M_{\infty}\mid\mathcal{F}_{k}]$ a.e.; and
$\left\Vert M\right\Vert =\left\Vert U\right\Vert +\left\Vert S\right\Vert $.
