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scjiang
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- TL;DR Summary
- 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 :)
$$
x_t = Ax_{t - 1} + \mathcal{N}(0, \sigma^2)
$$
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 :)