Complete moment convergence for the dependent linear processes with application to the state observers of linear-time-invariant systems

Let $X_t=\sum_{j=-\infty}^{\infty}A_j\varepsilon_{t-j}$ be a dependent linear process, where the $\{\varepsilon_n,\, n\in \mathbf{Z}\}$ is a sequence of zero mean $m$-extended negatively dependent ($m$-END, for short) random variables which is stochastically dominated by a random variable $\varepsilon$, and $\{A_n,\, n\in \mathbf{Z}\}$ is also a sequence of zero mean $m$-END random variables. Under some suitable conditions, the complete moment convergence for the dependent linear processes is established. In particular, the sufficient conditions of the complete moment convergence are provided. As an application, we further study the convergence of the state observers of linear-time-invariant systems.