Base Knowledge:
Let’s define the mean function of $x$ and $y$ and the variance function of $x$ as $m_{x}$ and $R_{x} (t, s)$ : Then we have that: So:
For $m_{y}$ we have:
Let’s now take into consideration the covariance function $R_{x} (t + τ, t)$ Definition: Substitute $x (t + τ) = A x (t + τ - 1) + \dots$ and $m_{x} (t + τ) = A m_{x} (t + τ - 1) + \dots$ , then simplify : Being $x (t)$ and $w (t)$ independent processes we have: Since $w (t)$ is white, so we have that $w (t + τ)$ and $w (t)$ are uncorrelated $\forall τ \in R$ we can say: Where:

$x (t) - m_{x} (t)$ only depends on the values of $w (i)$ for $i \leq t - 1$ (more over it’s a linear combination of $w (i)$ for $i \leq t - 1$ ). Then we can iterate an obtain: So: Now let’s see what $R_{x} (t, t)$ is equal to, starting from the definition: Like before we substitute $x (t + 1)$ and $m_{x} (t + 1)$ : Explicit form: Since of $x (t)$ and $w (t)$ are uncorrelated, like we said before, $x (t) - m_{x} (t)$ only depends on the values of $w (i)$ for $i \leq t - 1$ , so:

= E [(A x (t) - m_{x} (t)) \cdot (A x (t) - m_{x} (t))^{T}] + E [G w (t) w^{T} (t) G^{T}]

Where:

$E [(A x (t) - m_{x} (t)) \cdot (A x (t) - m_{x} (t))^{T}] = A R_{x} (t, t) A^{T}$
$E [G w (t) w^{T} (t) G^{T}] ≜ G Q G^{T}$ For simplicity we call $R (t, t) = P (t + 1)$ and declare the Discrete Lyapunov equation as: It tells us how the variance $P (t)$ of $x (t)$ evolves in time, $x (t)$ is not stationary. Finally we can conclude:

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