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A nonlinear spectral analyzer, called the L p-norm periodogram, is obtained by replacing the least-squares criterion with an L p-norm criterion in the regression formulation of the ordinary periodogram. In this paper, we study the statistical properties of the L p-norm periodogram for time series with continuous and mixed spectra. We derive the asymptotic distribution of the L p-norm periodogram and discover an important relationship with the so-called fractional autocorrelation spectrum that can be viewed as an alternative to the power spectrum in representing the serial dependence of a random process in the frequency domain. In comparison with the ordinary periodogram (p = 2), we show that by varying the value of p in the interval (1,2) the L p-norm periodogram can strike a balance between robustness against heavy-tailed noise, efficiency under regular conditions, and spectral leakage for time series with mixed spectra. We also show that the L p-norm periodogram can detect serial dependence of uncorrelated non-Gaussian time series that cannot be detected by the ordinary periodogram.