  What is the right statistics for the measurements of arrival times of a
quantum particle? Although this question is very old, it is still open. Usual
experiments are performed in far-field regime and this question becomes
unimportant, as a semiclassical analysis suffices. Nevertheless, the
development in the detector technology will soon allow for near-field
investigations, thus a better understanding of arrival time measurements is
needed. Since outcomes of quantum measurements are always described by POVMs,
various arrival time POVMs have been proposed. On the other hand many
physicists would agree that the arrival time statistics is given by the quantum
flux. This urges the question whether a POVM exists, which agrees approximately
with the quantum flux values on a reasonable set of wave functions. We answer
this question negatively for a very natural set of wave functions, but we
remark that the answer is very sensitive to the choice of the set, and provide
evidence for the existence of a POVM that agrees with the quantum flux on a
more restrictive set.
