On the large deviation principle for a quadratic functional of the autoregressive process
Let \(\{\gamma_ k\}_{k \in N}\) be independent identically distributed \(N(0,1)\) random variables and \(X_ k\) be an autoregressive process: \(X_ 0=0\), \(X_{k+1}= aX_ k+\gamma_{k+1}\). The large deviation principle for the quadratic functional \(S_ n ={1 \over n} \sum^ n_{k=1}X^ 2_ k\) is proved and the explicit rate function is given. The authors prove the following theorem. If \(-1<a<1\), then \(\{S_ n\}\) satisfies the large deviation principle with the rate function given by \[ I(x)=\begin{cases} -{1 \over 2} \ln \left[ {2x \over 1+ \sqrt{4a^ 2x^ 2+1}} \right] +{1 \over 2} [a^ 2+1]x-{1 \over 2} \sqrt{4a^ 2x^ 2+1} \quad & \text{for } x>0, \\ \infty & \text{for } x \leq 0. \end{cases} . \]
scientific article; zbMATH DE number 445309
scientific article