Strong law for submartingales with bounded increments (${\mathrm{liminf}}_n{Y}_n/n\ge 0\text{ a.s.}$)

If a submartingale has uniformly bounded increments, then ${\mathrm{liminf}}_n{Y}_n/n\ge 0$ almost surely. In its Doob decomposition in discrete time, the predictable compensator is nondecreasing, while the martingale part has bounded increments. The strong law for martingales with bounded increments makes the latter negligible on the linear scale.