  A practical constraint that comes in the way of spectrum estimation of a
continuous time stationary stochastic process is the minimum separation between
successively observed samples of the process. When the underlying process is
not band-limited, sampling at any uniform rate leads to aliasing, while certain
stochastic sampling schemes, including Poisson process sampling, are rendered
infeasible by the constraint of minimum separation. It is shown in this paper
that, subject to this constraint, no point process sampling scheme is
alias-free for the class of all spectra. It turns out that point process
sampling under this constraint can be alias-free for band-limited spectra.
However, the usual construction of a consistent spectrum estimator does not
work in such a case. Simulations indicate that a commonly used estimator, which
is consistent in the absence of this constraint, performs poorly when the
constraint is present. These results should help practitioners in rationalizing
their expectations from point process sampling as far as spectrum estimation is
concerned, and motivate researchers to look for appropriate estimators of
bandlimited spectra.
