地球統計學的研究者經常須要觀察資料的二階相依性，雖然半偏差較常被應用於地球統計資料分析上，但其實共變異量和半偏差在大部份的應用上是等價的，而且在評斷資料二階相依性的結構是否滿足可分離性時便必須利用共變異量。而在很多相關的統計推論比如可分離性檢定都必須知道樣本共變異量的抽樣分佈。然而在空間統計學的文獻上幾乎沒有關於樣本共變異量的研究。在此篇論文中我們證明在一個大部份地球統計模型都滿足的混合條件下，樣本共變異量的聯合極限分佈為多變量常態。文中我們先推導在任意隨機場上，樣本共變異量的機率分佈，然後探討在高斯隨機場上，定理所須的條件及結果都極其簡化。而一般地球統計資料分析也經常假設在高斯隨機場上，以此篇論文具有相當的應用價值。
Practitioners of geostatistics often need to investigate the second-order dependence structure of their data. Although the semivariogram is used more often in geostattistlcs, however, the covariogram is equivalent in characterizing the spatial dependence and it needs using the covariogram when assesssing whether the underlying process is separable. Hence, many statistical inferences such as testing the separability of the data require knowing the distributions of the sample covariogram. Unfortunately, there is almost no research about the sample covariogram in the spatial statistical literature. In this paper, we derive the asymptotic normality of the sample covariogram under a mixing condition which is satisfied by most geostatistical models. The distributional result is established for nongaussian random fields first and then specialized to gaussian random fields. The distributional results for cases of gaussian random fields are easy to use and the gaussian assumption is satisfied most of time in geostatistical practice. Thus, the results of this paper will be very useful for geostatistical data analysts.
