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The power spectral density of a continuous time wide sense stationary
stochastic process is the Fourier transform of its autocovariance
function. A common nonparametric estimator of this function is a smoothed
version of the periodogram (the Fourier transform of the sampled
autocovariance function), which is based on uniformly spaced samples of
the continuous time process. It is well known that this estimator is
inconsistent when the underlying power spectral density does not have
support restrcited to an suitable interval. For this reason, alternative
estimators based on irregularly spaced samples have been proposed. The
most common of these estimators is based on sampling according to a
Poisson process. Even though Poisson and other stochastic sampling schemes
involve somewhat high-tech theory, these are generally more difficult to
implement than regular sampling. In this talk, we shall show that one does
not necessarily have to use stochastic sampling for consistent estimation
of the power spectral density. For this purpose, we would use `shrinking
asymptotics', where the rate of uniform sampling is allowed to go to
infinity at a suitable rate as the sample size goes to infinity. We shall
also look into the issues of optimal rate, comparison with stochastic
sampling based estimators, extension to multivariate time series and
limiting distributions.
A possible criticism of `shrinking asymptotics' is that the inter-sample
distance cannot go to zero in real life sampling systems. For the problem
of estimation under the constraint of a minimum separation between
successive samples, we investigate the possibility of consistent
estimation through stochastic sampling, and come across some surprising
results. These findings are expected to lead to a more complete
understanding of the worth of stochastic sampling.
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