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摘要翻译：
定义并研究了一种基于增量比(IR)的与增量序列过零点有关的统计量，用于测量随机路径的粗糙度。这种统计量的主要优点是对平滑加性和乘性趋势的鲁棒性以及对无穷方差过程的适用性。IR统计极限（以下称为IR粗糙度）的存在与切过程的存在密切相关。考虑了红外粗糙度存在并被显式计算的三种特殊情况。首先，对于具有光滑扩散和漂移系数的扩散过程，其IR粗糙度与布朗运动的IR粗糙度一致，得到了其收敛速度。其次，在不要求平稳性条件的一般假设下，详细研究了粗糙高斯过程的情形。第三，建立了具有α-α稳定切线过程的L′eVy过程的IR粗糙度，并利用中心极限定理估计了(0，2)$中的分数阶参数α-α。
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英文标题：
《Measuring the roughness of random paths by increment ratios》
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作者：
Jean-Marc Bardet (SAMM), Donatas Surgailis
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最新提交年份：
2010
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分类信息：
一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a L\'evy process with $\alpha-$stable tangent process is established and can be used to estimate the fractional parameter $\alpha \in (0,2)$ following a central limit theorem.
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PDF链接：
https://arxiv.org/pdf/802.0489
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