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 $X$.

翻译：我们考虑一类具有有限对称且绝对连续控制测度的平稳实可调和对称$α$-稳态过程$X=\left\{X(t):t\in\mathbb{R}\right\}$的非遍历性族。将控制测度的密度函数称为$X$的谱密度。此类过程具备LePage级数表示形式且为条件高斯过程，由此可推导$X$样本函数的非遍历极限。特别地，我们分别给出了$X$的经验特征函数及滞后过程$\left\{X(t+h)-X(t):t\in\mathbb{R}\right\}$（其中$h>0$）的非遍历极限显式表达式。该过程可等价表示为具有随机频率的正弦波级数，这些频率独立同分布于以$X$的（归一化）谱密度为概率密度函数的分布。基于利用周期图实现的强相合频率估计，我们提出了一个强相合的谱密度估计量。该方法周期图计算高效快捷，且不受$X$非遍历性的影响。