论文标题
固定的真实对称对称$α$稳定过程的非共性统计和光谱密度估计
Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric $α$-stable processes
论文作者
论文摘要
我们考虑使用有限的对称且绝对连续的控制措施，我们考虑使用有限的和绝对连续的控制措施的非共和对称$α$稳定过程$ x = \ left \ {x（x（t）：t \ in \ mathbb {r} \ right \} $。我们将其密度函数称为$ x $的光谱密度。这些过程允许使用lepage系列表示，并且有条件地是高斯，这使我们能够在$ x $上得出样本函数的非共性限制。特别是，我们对$ x $的经验特征函数的非共性限制和滞后过程$ \ weft \ {x（t+h）-x（t）：t \ in \ in \ mathbb {r} \ right \} $，分别为$ h> 0 $。该过程将等效表示为一系列正弦波，其随机频率为I.I.D。以$ x $的（归一化）光谱密度为其概率密度函数。基于使用周期图的强频估计，我们提出了光谱密度的强估计器。周期图的计算快速有效，我们的方法不受$ x $的非效率的影响。
We consider non-ergodic class of stationary real harmonizable symmetric $α$-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 $X$.
