  In any Bell test, loopholes can cause issues in the interpretation of the
results, since an apparent violation of the inequality may not correspond to a
violation of local realism. An important example is the coincidence-time
loophole that arises when detector settings might influence the time when
detection will occur. This effect can be observed in many experiments where
measurement outcomes are to be compared between remote stations because the
interpretation of an ostensible Bell violation strongly depends on the method
used to decide coincidence. The coincidence-time loophole has previously been
studied for the Clauser-Horne-Shimony-Holt (CHSH) and Clauser-Horne (CH)
inequalities, but recent experiments have shown the need for a generalization.
Here, we study the generalized "chained" inequality by Pearle-Braunstein-Caves
(PBC) with two or more settings per observer. This inequality has applications
in, for instance, Quantum Key Distribution where it has been used to
re-establish security. In this paper we give the minimum coincidence
probability for the PBC inequality for all N and show that this bound is tight
for a violation free of the fair-coincidence assumption. Thus, if an experiment
has a coincidence probability exceeding the critical value derived here, the
coincidence-time loophole is eliminated.
