Autoregressive time series are \(L_p\)-mixingales (Q1775364)

Let \((Y_n)_{n \in \mathbb{Z}}\) and \((X_n)_{n \in \mathbb{Z}}\) be two sequences of real random variables such that \[ X_n = \sum_{i=1}^\infty d_i X_{n-i} + Y_n \quad \text{a.s.,} \quad n \in \mathbb{Z}, \] where the \(d_i\) are real constants such that \(\sum_{i=1}^\infty | d_i| < \infty\). Suppose in addition that there is \(p \geq 1\) such that \(\limsup_{n\to -\infty} \| X_n\| _p < \infty\), \(\sup_{n\in\mathbb{Z}} \| Y_n\| _p <\infty\), and such that \(Z_n := Y_n - E(Y_n)\) is an \(L_p\)-mixingale with respect to some filtration \((\mathcal{F}_n)_{n\in\mathbb{Z}}\), i.e., there are non-negative constants \(C_n^Y\) und \(\xi_m^Y\) such that \[ \| E (Z_n \mid \mathcal{F}_{n-m}) \| _p \leq C_n^Y \xi_m^Y\quad\text{and}\quad\| Z_n - E(Z_n \mid \mathcal{F}_{n+m})\| _p \leq C_n^Y \xi_{m+1}^Y \] for all \(n\in\mathbb{Z}\), \(m\in\mathbb{N}_0\), and it holds \(\lim_{m\to\infty} \xi_m = 0\). It is shown that, provided \((C_n^Y)_{n\in \mathbb{Z}}\) is bounded, \((X_n - E(X_n))_{n\in\mathbb{Z}}\) is an \(L_p\)-mixingale with respect to the same filtration, and that \(C_n^X\) can be taken to be 1. Under further conditions on the decay of \((d_n)_{n\in\mathbb{N}}\) and \((\xi_m^Y)_{m \in \mathbb{N}}\) it is shown that \((\xi_m^X)_{m\in\mathbb{N}}\) decays similarly. Finally, a sufficient condition for \((X_n)_{n \in \mathbb{N}}\) to satisfy the strong law of large numbers is presented.