I figured it out (mostly).
Suppose we have a Poisson (or other) process whose mean slowly drifts in time. (Slow: on a time scale much longer than the time between samples) by a amount small in comparison to the nominal mean. As a PERIODIC model:
LocalMeanOft = NominMean + dM cos om t
If I histogrammed the data accumulated for a time much longer than 2 Pi/om I would find a mean of NominMean. If I histogrammed the data accumulated over a time SHORT in comparison to 2 Pi/om, I would find a mean which depended on where in the cycle the data was collected.
(i) because dM << NominMean, the effect on the breadth of the distribution is small.
(ii) the 'pile-up' of data points at the nominal mean simply reflects the fact that the local mean (in the periodic MODEL above) is as likely to
be larger than the nominal mean as to be smaller. This means that if the TOTAL sampling time is short in comparison to 2 Pi/om, I will see a
well-behaved Poisson histogram. The longer the sampling time, the more pronounced the `anomalous' peak in the histogram will be.
Issue: because so many counts pile up at mean, it will somewhat distort a fits. So the real questions are
(1) How to get the best signal/noise out of fits. This probably entails identifying the longest time for which the peak pile up is (almost) NOT present. This is probably encoded in the autocorrelation function. Perhaps moving averages are a better guide, to remove SHORT-duration (high frequency) noise. I probably need to (i) break up the data for the TOTAL sampling time into intervals in which rate is reasonably constant (ii) Fit these as independent sets, to max S/N ratio (minimize effect of pile-up of counts at mean)
(2) Perhaps for each interval subtract fit from data, then examine the t-dependence of the difference.
These issues must be completely understood by people who do time-series analysis. Can anyone point me to some references on this?
Again, thanks Daniel.