Nonnegative martingale
= Nonnegative martingale
{title2=$M_n\geq0$}
A nonnegative <martingale> takes nonnegative values <almost surely> at each time. In discrete time its <expectations> are constant, so it is bounded in $L^1$ and the <Martingale convergence theorem> gives a finite integrable almost sure limit. Its <expectations> need not converge to the <expectation> of that limit; <uniform integrability> is needed for <convergence in L1>.