高斯序列二次变差极限定理成立的充分和必要条件
Sufficient and Necessary Conditions for Limit Theorems for Quadratic Variations of Gaussian Sequences
摘要：
高斯过程的二次变分在随机分析和模型参数估计等应用中起着重要的作用，因此该主题在文献中得到了广泛的研究。本文研究了一般高斯过程的收敛问题，给出了依概率收敛、几乎必然收敛、$L^p$ -收敛和依律收敛的充分条件和必要条件.此外，我们研究了一般高斯向量，从不同的有趣的情况下，包括一阶或二阶二次变化，可以通过适当的选择的基础向量进行研究。最后，我们提供了一个实用和简单的方法来解决这个问题，大大简化了现有的方法。
英文摘要：
Quadratic variations of Gaussian processes play important role in both stochastic analysis and in applications such as estimation of model parameters, and for this reason the topic has been extensively studied in the literature. In this article we study the problem for general Gaussian processes and we provide sufficient and necessary conditions for different types of convergence which include convergence in probability, almost sure convergence, $L^p$-convergence as well as convergence in law. Furthermore, we study general Gaussian vectors from which different interesting cases including first or second order quadratic variations can be studied by appropriate choice of the underlying vector. Finally, we provide a practical and simple approach to attack the problem which simplifies the existing methodology considerably.
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