Large and moderate deviations for moving average processes
Let \(\{\omega_i\}\) be iid random variables. Define \(X_k = \sum_{i=-\infty}^{\infty} a_{i+k}\omega_i\), \(S_n=\sum_{k=1}^n X_k\). Assume that the following conditions are satisfied: (H1) \(\sum a_i^2<\infty\), \(E\omega_i=0\), \(E\omega_i^2=1\), \(|\omega_i|\leq C\) where \(C\) is a constant; (H2) the spectral density of \(\{X_t\}\) is continuous at 0 and square integrable. The authors prove that then \(S_n/n\) satisfies a large deviation principle with speed \(n\). If \(\{b_n\}\) is a sequence such that \(b_n\to \infty\), \(b_n/\sqrt n\to 0\), then \(S_n/(b_n\sqrt n)\) also satisfies a large deviation principle, this time with the speed \(b_n^2\). In the both cases the good rate functions are given.
scientific article; zbMATH DE number 1796102
scientific article