Consider a stationary real harmonic symmetric $\alpha$-stable process $X=\left\{X

The stationary real harmonic symmetric $\alpha$ stable process $X$ is conditionally Gaussian that accepts the LePage series representation and allows us to derive the non-ergodic limit of the sample function on $X$. In particular, the non-ergodic limit of the empirical characteristic function $X$ and the lag process $\left\{X(t+h)-X

This process allows an equivalent representation as a series of sinusoids with random frequencies whose probability density function is actually the (normalized) spectral density of X. We present a strong and consistent estimator of spectral density based on a strong and consistent frequency estimate using the periodogram. Computing the periodogram is fast and efficient, and the method does not suffer from the non-ergodicity of $X$. Most notably, no prior knowledge of process parameters such as the stability index $\alpha$ is required.

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