= Almost-sure martingale convergence to a nonintegrable limit
Let independent increments $\xi_k$ equal $4^k$ or $-4^k$ with probabilities $2^{-k-1}$ each, and zero otherwise. Every partial sum $M_n=\sum_{k\leq n}\xi_k$ is integrable and is a <martingale>. The <Borel-Cantelli lemma> makes only finitely many increments nonzero almost surely, so $M_n$ has a finite almost-sure limit $Y$. Yet $\mathbb E|Y|=\infty$: the disjoint events that only the $k$th increment is nonzero have probabilities at least $c2^{-k}$ for $c=\prod_j(1-2^{-j})>0$, and contribute at least $c2^k$ each. This does not contradict the <martingale convergence theorem>, whose integrability conclusion requires a uniform first-moment bound.
// Target: markov-process.bigb
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