高维自协方差矩阵的大样本性质
Large sample behaviour of high dimensional autocovariance matrices
摘要：
当观测值来自无穷维向量线性过程且对系数矩阵作适当（强）假设时，已知$\hatΓ_u+\hatΓ_u^*$（$u$阶样本自协方差矩阵$\hatΓ_u$的对称和）极限谱分布（LSD）的存在性.在较弱的条件下，统一地证明了$\hatΓ_u+\hatΓ_u^*$，$\hatΓ_u\hatΓ_u^*$，$\hatΓ_u\hatΓ_u^*+\hatΓ_k\hatΓ_k^*$等矩阵中任意对称多项式的LSD都存在.我们的方法是通过更直观的代数方法的自由概率结合矩量法。因此，我们能够提供一个一般的描述的限制，在一些自由的自变量。前面的结果都是特例。我们建议统计使用这些LSD和相关结果的顺序确定和白噪声测试。
英文摘要：
The existence of limiting spectral distribution (LSD) of $\hatΓ_u+\hatΓ_u^*$, the symmetric sum of the sample autocovariance matrix $\hatΓ_u$ of order $u$, is known when the observations are from an infinite dimensional vector linear process with appropriate (strong) assumptions on the coefficient matrices. Under significantly weaker conditions, we prove, in a unified way, that the LSD of any symmetric polynomial in these matrices such as $\hatΓ_u+\hatΓ_u^*$, $\hatΓ_u\hatΓ_u^*$, $\hatΓ_u\hatΓ_u^*+\hatΓ_k\hatΓ_k^*$ exist. Our approach is through the more intuitive algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. All the previous results follow as special cases. We suggest statistical uses of these LSD and related results in order determination and white noise testing.
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。
