随机场多重分形积的极限定理
Limit theorems for multifractal products of random fields
摘要：
研究了随机场多重分形积的渐近性质。所得到的极限定理提供了累积域在空间$L_q.$中收敛的充分条件。给出了累积域的收敛速率的新结果。给出了极限定理和Rényi函数计算的简单统一条件。它们比已知的一维结果中的限制性更小。开发的方法也适用于多维多重分形措施。最后给出了一类新的几何$φ$ -亚高斯随机场的例子。在这种情况下，一般假设具有简单的形式，并且可以仅用协方差函数表示。
英文摘要：
This paper investigates asymptotic properties of multifractal products of random fields. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces $L_q.$ New results on the rate of convergence of cumulative fields are presented. Simple unified conditions for the limit theorems and the calculation of the Rényi function are given. They are less restrictive than those in the known one-dimensional results. The developed methodology is also applied to multidimensional multifractal measures. Finally, a new class of examples of geometric $φ$-sub-Gaussian random fields is presented. In this case, the general assumptions have a simple form and can be expressed in terms of covariance functions only.
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