多重分形分析中的线性化效应：来自随机能量模型的启示
Linearization effect in multifractal analysis: Insights from the Random Energy Model
摘要：
多重分形分析中的线性化效应的分析，因此，多重分形过程的矩的估计，是重新审视借用的概念，从统计物理学的无序系统，特别是从所谓的随机能量模型的分析。考虑一个标准的多重分形过程（复合泊松运动），选择作为一个简单的代表性例子，我们表明：i）存在一个临界阶$q^*$超过这一时刻，虽然有限，不能通过经验平均值估计，无论样本大小的观察; ii）多重分形指数必然表现为线性$q$，为$q > q^*$。泰勒的随机能量模型进行的分析，复合泊松运动，我们提供了明确的和定量的预测值的$q^*$和斜率控制的线性行为的多重分形指数。这些数量被证明是只与定义的多重分形过程，而不是依赖于样本大小的观察。蒙特卡罗模拟，进行了大量的大样本量实现的复合泊松运动，舒适和扩展这些分析。
英文摘要：
The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the analysis of the so-called Random Energy Model. Considering a standard multifractal process (compound Poisson motion), chosen as a simple representative example, we show: i) the existence of a critical order $q^*$ beyond which moments, though finite, cannot be estimated through empirical averages, irrespective of the sample size of the observation; ii) that multifractal exponents necessarily behave linearly in $q$, for $q > q^*$. Tayloring the analysis conducted for the Random Energy Model to that of Compound Poisson motion, we provide explicative and quantitative predictions for the values of $q^*$ and for the slope controlling the linear behavior of the multifractal exponents. These quantities are shown to be related only to the definition of the multifractal process and not to depend on the sample size of the observation. Monte-Carlo simulations, conducted over a large number of large sample size realizations of compound Poisson motion, comfort and extend these analyses.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
