Deterministic summable drift correction ($Z_n=X_n+{\sum }_{j\ge n}{\delta }_j$)

When the drift bound is a deterministic summable sequence ${\delta }_n\ge 0$, the displayed correction is a nonnegative supermartingale. Its added tail tends to zero, so the almost sure supermartingale convergence theorem directly implies finite almost sure convergence of $X_n$. This simpler tail correction requires deterministic bounds; an arbitrary future random drift tail is not generally measurable at the current time.