It is well known that if the power spectral density of a continuous time stationary stochastic process does not have a compact support, data sampled from that process at any uniform sampling rate leads to biased and inconsistent spectrum estimators. In a recent paper, the authors showed that the smoothed periodogram estimator can be consistent, if the sampling interval is allowed to shrink to zero at a suitable rate as the sample size goes to infinity. In this paper, this `shrinking asymptotics' approach is used to obtain the limiting distribution of the smoothed periodogram estimator of spectra and cross-spectra. It is shown that, under suitable conditions, the scaling that ensures weak convergence of the estimator to a limiting normal random vector can range from cube-root of the sample size to square-root of the sample size, depending on the strength of the assumption made. The results are used to construct asymptotic confidence intervals for spectra and cross spectra. It is shown through a Monte-Carlo simulation study that these intervals have appropriate empirical coverage probabilities at moderate sample sizes.