Proposition: Arrival discontinuity

A deterministic arrival time $t_n\to t$, $t_n\ne t$, need not converge in total variation as a classical time record. A fixed deadline can also have discontinuous acceptance probability.

Proof. The point masses ${\delta }_{t_n}$ and ${\delta }_t$ have total variation distance one. If $t_n=t+1/n$, the event $(-\infty ,t]$ has probabilities zero and one, respectively.