Abstract We describe two classes of Gaussian multi-self-similar random fields: with strictly stationary rectangular increments and with mild stationary rectangular increments. We find explicit spectral and moving average representations for the fields with strictly stationary rectangular increments and characterize fields with mild stationary rectangular increments by the properties of covariance functions of their Lamperti transformations as well as in terms of their spectral densities. We establish that both classes contain not only fractional Brownian sheets and we provide corresponding examples. As a by-product, we obtain a new spectral representation for the fractional Brownian motion.

中文翻译：

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具有不同矩形增量平稳特性的高斯多重自相似随机场

摘要 我们描述了两类高斯多重自相似随机场：严格平稳矩形增量和温和平稳矩形增量。我们找到了具有严格平稳矩形增量的场的显式谱和移动平均表示，并通过它们的 Lamperti 变换的协方差函数的性质以及它们的谱密度来表征具有温和平稳矩形增量的场。我们确定这两个类不仅包含分数布朗表，并且我们提供了相应的示例。作为副产品，我们获得了分数布朗运动的新谱表示。