 Finite-jump boundary crossing identity

Let $U_1,\dots ,U_n$ be independent uniform random variables on $[0,1]$, let ${\theta }_i>0$, and set $S={\sum }_i{\theta }_i\in (0,1)$ and $X_t={\sum }_i{\theta }_i{1}_{\{U_i\le t\}}$. ThenIndeed, the complementary crossing is that of level one by the uniform-arrival residual mass martingale $M_t=(S-X_t)/(1-t)$, with $M_0=S$. Its jumps are downward, so its first crossing of one is continuous. An arrival coincides with a possible crossing time with probability zero: the possible times belong to the finite deterministic set obtained from subsets of the weights. Stop at that crossing or at $r<1$. The stopped values are in $[0,1]$, and as $r\uparrow 1$ they converge to the crossing indicator because all arrivals occur before one almost surely. The optional stopping theorem and dominated convergence theorem give crossing probability $S$. For $S=1$, the strict displayed inequality already fails at time zero.