具有高斯新息的AR（1）和MA（1）过程大偏差的显式二元速率函数
Explicit Bivariate Rate Functions for Large Deviations in AR(1) and MA(1) Processes with Gaussian Innovations
摘要：
通过给出随机向量序列$(\boldsymbol{S}_n)_{n \in \N} = \left(n^{-1}(\sum_{k=1}^n X_k, \sum_{k=1}^n X_k^2)\right)_{n \in \N}$的二元速率函数，研究了具有独立高斯新息的中心平稳AR（1）和MA（1）过程的大偏差性质.在AR（1）情形下，我们也给出了二元随机序列$(\W_n)_{n \geq 2} = \left(n^{-1}(\sum_{k=1}^n X_k^2, \sum_{k=2}^n X_k X_{k+1})\right)_{n \geq 2}$的显式速率函数。利用压缩原理，我们还给出了序列$(n^{-1} \sum_{k=1}^n X_k)_{n \in \N}$、$(n^{-1} \sum_{k=1}^n X_k^2)_{n \geq 2}$和$(n^{-1} \sum_{k=2}^n X_k X_{k+1})_{n \geq 2}$的显式速率函数.在AR（1）情形下，我们给出了Yule-Walker估计的显式偏差函数的一个已知结果的新证明。
英文摘要：
We investigate large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate functions for the sequence of random vectors $(\boldsymbol{S}_n)_{n \in \N} = \left(n^{-1}(\sum_{k=1}^n X_k, \sum_{k=1}^n X_k^2)\right)_{n \in \N}$. In the AR(1) case, we also give the explicit rate function for the bivariate random sequence $(\W_n)_{n \geq 2} = \left(n^{-1}(\sum_{k=1}^n X_k^2, \sum_{k=2}^n X_k X_{k+1})\right)_{n \geq 2}$. Via Contraction Principle, we provide explicit rate functions for the sequences $(n^{-1} \sum_{k=1}^n X_k)_{n \in \N}$, $(n^{-1} \sum_{k=1}^n X_k^2)_{n \geq 2}$ and $(n^{-1} \sum_{k=2}^n X_k X_{k+1})_{n \geq 2}$, as well. In the AR(1) case, we present a new proof for an already known result on the explicit deviation function for the Yule-Walker estimator.
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