= Uniform-arrival residual mass martingale
{title2=$M_t=\sum_i\theta_i\mathbf1_{\{U_i>t\}}/(1-t)$}
For finitely many independent uniform arrival times on $[0,1]$, and fixed positive weights, normalize the unarrived mass by the remaining time. Conditional on not having arrived by $s$, each arrival remains uniform on $(s,1)$, giving the <martingale> identity. The process is <càdlàg>, nonnegative, and has constant mean equal to the total weight. It vanishes eventually before time one on each typical path, so it is not <uniformly integrable> when the total weight is positive.
Back to article page
