Definition:Linear Filter/Stable

Definition
Let $S$ be a stationary stochastic process governed by a white noise process:


 * $\map z t = \mu + a_t$

where:
 * $\mu$ is a constant mean level
 * $a_t$ is an independent shock at timestamp $t$.

Let $L$ be a linear filter on $S$:

Consider the sequence $\sequence {\psi k}$ formed by the weight function $\psi$ of $L$.

Suppose that:
 * $\displaystyle \sum_k \size {\psi_k} < \infty$

then $L$ is said to be stable, and the model for $S$ is stationary.

Hence $\mu$ is the mean about which $S$ varies.