  We consider non-ergodic class of stationary real harmonizable symmetric
$\alpha$-stable processes $X=\left\{X(t):t\in\mathbb{R}\right\}$ with a finite
symmetric and absolutely continuous control measure. We refer to its density
function as the spectral density of $X$. These processes admit a LePage series
representation and are conditionally Gaussian, which allows us to derive the
non-ergodic limit of sample functions on $X$. In particular, we give an
explicit expression for the non-ergodic limits of the empirical characteristic
function of $X$ and the lag process $\left\{X(t+h)-X(t):t\in\mathbb{R}\right\}$
with $h>0$, respectively. The process admits an equivalent representation as a
series of sinusoidal waves with random frequencies which are i.i.d. with the
(normalized) spectral density of $X$ as their probability density function.
Based on strongly consistent frequency estimation using the periodogram we
present a strongly consistent estimator of the spectral density. The
periodogram's computation is fast and efficient, and our method is not affected
by the non-ergodicity of $X$.
