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Please help me with the following exercise from probability theory.

With $n \geq 0$, let $X_{n}$ be a nonnegative process with Doob decomposition $X_{n} = M_{n} + A_{n}$ where $M_{n}$ is the martingale part and $A_n$ the predictable part. Set $a_{k}=A_{k}-A_{k-1}$ and let $a_{k}^{+} = \max(a_{k},0)$.

Assume $E[\sum_{k \geq 1} a_{k}^{+}]< \infty$, prove that $X_{n}$ converges a.s. to a finite limit.

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Assume all random variables are in $L^1$, i.e., absolutely integrable. We do not need $(A_n)$ to be predictable.

Using $X_n\geq 0$, we get $M_n^-=\max(0,A_n-X_n)\leq A_n^+\leq A_0^+ +\sum_{k\geq 1} a_k^+.$ Taking expectations shows that $\mathbb{E}(M_n^-)\leq \mathbb{E}\left(A_0^+ +\sum_{k\geq 1} a_k^+\right)<\infty.$ Since $(M_n)$ is a martingale, we have
$\mathbb{E}(|M_n|)= \mathbb{E}(M_n)+2 \mathbb{E}(M_n^-) = \mathbb{E}(M_0)+2 \mathbb{E}(M_n^-), $ which combined with the previous bound, shows that $(M_n)$ is bounded in $L^1$. By the martingale convergence theorem, $M_n\to M_\infty$ almost surely. By Fatou's Lemma, $M_\infty$ is also in $L^1$.

Turning to the $(A_n)$ process, for $n\geq m$ we have $A_n\leq A_m+\sum_{k>m} a_k^+,$ so that $\sup_{n\geq m}\, A_n\leq A_m+\sum_{k>m} a_k^+.$ In particular, the random variable on the left is almost surely finite. Taking the liminf in $m$ on both sides, we get $\limsup_m\, A_m\leq \liminf_m\ A_m.$

This shows that $A_m\to A_\infty$ in $[-\infty,\infty]$. We already know that $A_m$ is almost surely bounded above, so $A_\infty<\infty$. Letting $m\to\infty$ in $A_m=X_m-M_m\geq -M_m$ gives $A_\infty\geq -M_\infty >-\infty$, and the proof is complete.