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维度与样本量之比趋零时样本自协方差矩阵的联合收敛及应用AI 解读

Consider a high-dimensional linear time series model where the dimension $p$ and the sample size $n$ grow in such a way that $p/n\to 0$. Let $\hat{\Gamma }_{u}$ be the $u$th order sample autocovariance matrix. We first show that the LSD of any symmetric polynomial in $\{\hat{\Gamma }_{u},\hat{\Gamma }_{u}^{*},u\geq 0\}$ exists under independence and moment assumptions on the driving sequence together with weak assumptions on the coefficient matrices. This LSD result, with…

高维自协方差矩阵的大样本行为AI 解读

The existence of limiting spectral distribution (LSD) of $\hat{\Gamma}_{u}+\hat{\Gamma}_{u}^{*}$, the symmetric sum of the sample autocovariance matrix $\hat{\Gamma}_{u}$ 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 $\hat{\Gamma}_{u}+\hat{\Gamma}_{u}^{*}$…

自回归中利用 Bootstrap 进行 Edgeworth 修正AI 解读

We prove that the distribution of least-squares estimates in autoregressions can be bootstrapped with accuracy $o(n^{-1/2})$ a.s., thereby improving the normal approximation error of $O(n^{-1/2})$.