I Relevant Literature for Noisy Linear Dynamical Systems

AI Thread Summary
The discussion focuses on the existence of time averages in noisy linear dynamical systems defined by the equation x_t = Ax_{t - 1} + N(0, σ²), where A is orthogonal. The main inquiry is whether the limit of the time average of a continuous function f(x_t, x_{t + 1}) exists and under what conditions related to f and A. The participant has explored ergodicity and random dynamical systems but finds it challenging to locate relevant literature, particularly since the system does not fit traditional Markov models. Recommendations for literature or textbooks that address these concepts are sought, particularly regarding the eigenvalues of A and their norms. The discussion emphasizes the need for clarity on the conditions that ensure the existence of the time average.
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Trying to compute the time average of a linear dynamical system with noise, but unable to find any relevant literature or keywords. Would deeply appreciate any guidance.
I have been attempting a question about noisy linear dynamical systems lately. Specifically, suppose we are given a linear dynamical system
$$
x_t = Ax_{t - 1} + \mathcal{N}(0, \sigma^2)
$$
where $A$ is orthogonal, $x_t \in \mathbb{R}^n$, and $\mathcal{N}(0, \sigma^2)$ is a normal distribution. Also, let $f$ be an arbitrary well-behaved function (say continuous) on $\mathbb{R}^n \times \mathbb{R}^n$. Does the time average
$$
\lim_{T \rightarrow \infty} \frac{1}T \sum_{t = 0}^{T - 1} f(x_t, x_{t + 1})
$$
exist, and if so, under which conditions on $f, A$?

I have tried reading about ergodicity and random dynamical systems, but am still struggling to find the right keywords and literature for this question. The system doesn't seem to be Markov (since $A$ is merely orthogonal, not a probability transition matrix), so haven't looked into the MC literature.

If anybody has any literature or textbook recommendations, it would be deeply appreciated :)
 
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I'm not 100%, and it depends on your choice of notm, but if the eigenvalues of ##A## are Real, you would want them to have norm in ##[-1,1]##.
 
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