The main object in the statistical analysis of high-frequency financial data are sums of functionals of increments of stochastic processes, and statistical inference is based on the asymptotic behaviour of these sums as the mesh of the observation times tends to zero. Inspired by the famous Hayashi–Yoshida estimator for the quadratic covariation based on two asynchronously observed stochastic processes, we investigate similar sums for general functionals. We find that our results differ from corresponding results for synchronous observations, a case which has been well studied in the literature, and we observe that the asymptotic behaviour in the setting of asynchronous observations is not only determined by the nature of the functional, but also depends crucially on the asymptotics of the observation scheme. Several examples are discussed, including the case of \(f(x_{1},x_{2}) = |x_{1}|^{p_{1}} |x_{2}|^{p_{2}}\) which has various applications in empirical finance.

中文翻译：

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林-吉田型泛函的大数定律

高频金融数据统计分析的主要对象是随机过程增量的函数和，而统计推断基于这些和的渐近行为，因为观察时间的网格趋于零。受著名的Hayashi-Yoshida估计器的启发，该估计器基于两个异步观察到的随机过程进行二次协方差，我们研究了一般函数的相似和。我们发现我们的结果与同步观测的相应结果有所不同，同步观测的情况在文献中已得到很好的研究，并且我们观察到异步观测设置中的渐近行为不仅取决于函数的性质，而且还取决于关键取决于观察方案的渐近性。讨论了几个例子，\（f（x_ {1}，x_ {2}）= | x_ {1} | ^ {p_ {1}} | x_ {2} | ^ {p_ {2}} \）在经验金融中有多种应用。