Almost sure submartingale convergence theorem (${\sup }_n\mathbb{E}{Y}_n^{+}<\infty ⟹Y_n\to Y_{\infty }\text{ a.s.}$)

A submartingale with ${\sup }_n\mathbb{E}{Y}_n^{+}<\infty$ converges almost surely to a finite integrable random variable. The positive-part bound also bounds negative parts because $\mathbb{E}{Y}_n\ge \mathbb{E}{Y}_0$. The result specializes to an $L^1$-bounded martingale, and hence to every nonnegative martingale. It does not by itself give convergence in L1; uniform integrability supplies that stronger conclusion.