Rosenblatt过程的变分和Hurst指数估计
Variations and Hurst index estimation for a Rosenblatt process using longer filters
摘要：
Rosenblatt过程是一个生活在二阶Wiener混沌中的自相似非高斯过程，在所谓的\textquotedblleft非中心极限定理\textquotedblright中作为相关随机序列的极限出现.它与分数布朗运动具有相同的协方差。我们研究了基于长滤波器的Rosenblatt过程的二次变化的渐近分布，包括基于高阶有限差分和基于小波的滤波器。我们找到确切的公式的限制分布，然后我们使用设计强一致的自相似参数$H$的估计。与分数布朗运动的情况不同，无论现在滤波器阶数多高，估计量都不会渐近正态，而是以均方收敛到Rosenblatt过程在时间1处的观测值。
英文摘要：
The Rosenblatt process is a self-similar non-Gaussian process which lives in second Wiener chaos, and occurs as the limit of correlated random sequences in so-called \textquotedblleft non-central limit theorems\textquotedblright. It shares the same covariance as fractional Brownian motion. We study the asymptotic distribution of the quadratic variations of the Rosenblatt process based on long filters, including filters based on high-order finite-difference and wavelet-based schemes. We find exact formulas for the limiting distributions, which we then use to devise strongly consistent estimators of the self-similarity parameter $H$. Unlike the case of fractional Brownian motion, no matter now high the filter orders are, the estimators are never asymptotically normal, converging instead in the mean square to the observed value of the Rosenblatt process at time 1.
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