  In the matter of selection of sample time points for the estimation of the
power spectral density of a continuous time stationary stochastic process,
irregular sampling schemes such as Poisson sampling are often preferred over
regular (uniform) sampling. A major reason for this preference is the
well-known problem of inconsistency of estimators based on regular sampling,
when the underlying power spectral density is not bandlimited. It is argued in
this paper that, in consideration of a large sample property like consistency,
it is natural to allow the sampling rate to go to infinity as the sample size
goes to infinity. Through appropriate asymptotic calculations under this
scenario, it is shown that the smoothed periodogram based on regularly spaced
data is a consistent estimator of the spectral density, even when the latter is
not band-limited. It transpires that, under similar assumptions, the estimators
based on uniformly sampled and Poisson-sampled data have about the same rate of
convergence. Apart from providing this reassuring message, the paper also gives
a guideline for practitioners regarding appropriate choice of the sampling
rate. Theoretical calculations for large samples and Monte-Carlo simulations
for small samples indicate that the smoothed periodogram based on uniformly
sampled data have less variance and more bias than its counterpart based on
Poisson sampled data.
