含噪声观测的调和稳定序列极小极大外推问题
Minimax Extrapolation Problem For Harmonizable Stable Sequences With Noise Observations
AI总结：
针对含噪声的调和对称α-稳定随机序列，研究泛函最优线性外推估计问题，分别在谱确定和谱不确定条件下推导了最优估计的误差、谱特征及最不利谱密度、极小极大谱特征的计算公式。
摘要：
我们研究泛函$A ξ=\sum_{j = 0}^{\infty} a_j ξ_j$的最优线性估计问题，该泛函依赖于随机序列$\{ξ_j,j\in\mathbb Z\}$的未知值$ξ_j,j=0,1,\dots$，观测值为序列$\{ξ_k+η_k,k\in\mathbb Z\}$在点$k = -1, -2, \dots$处的取值，其中$\{ξ_k,k\in\mathbb Z\}$和$\{η_k,k\in\mathbb Z\}$是相互独立的调和对称$α$-稳定随机序列，其谱密度$f(θ)>0$和$g(θ)>0$满足极小性条件。我们分别在谱确定条件和谱不确定条件下研究该问题。在谱确定条件下（即序列的谱密度精确已知），推导了计算该泛函最优线性估计的误差值和谱特征的公式。在谱不确定情况下（即序列的谱密度不精确已知，但给定一类容许谱密度），推导了确定最不利谱密度和极小极大谱特征的关系式。
英文摘要：
We consider the problem of optimal linear estimation of the functional $A ξ~=~\sum_{j = 0}^{\infty} a_j ξ_j$ that depends on the unknown values $ξ_j,j=0,1,\dots, $ of a random sequence $\{ξ_j,j\in\mathbb Z\}$ from observations of the sequence $\{ξ_k+η_k,k\in\mathbb Z\}$ at points $k = -1, -2, \dots$, where $\{ξ_k,k\in\mathbb Z\}$ and $\{η_k,k\in\mathbb Z\}$ are mutually independent harmonizable symmetric $α$-stable random sequences which have the spectral densities $f(θ)>0$ and $g(θ)>0$ satisfying the minimality condition. The problem is investigated under the condition of spectral certainty as well as under the condition of spectral uncertainty. Formulas for calculating the value of the error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty where spectral densities of the sequences are exactly known. In the case of spectral uncertainty where spectral densities of the sequences are not exactly known, while a class of admissible spectral densities is given, relations that determine the least favorable spectral densities and the minimax spectral characteristic are derived.
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