Definition of an AR(1) process
In an Autoregressive Process, a time series can be generated based on a stochastic difference equation.
\begin{align}
X_t = c + \phi \, X_{t-1} + \epsilon
\end{align}
Typically, $\epsilon$ is chosen to be normally distributed and $0 < \phi < 1$.
What I already know
As described on Wikipedia, as well as on Stack Exchange, one can derive the mean and the variance of the resulting time series if one knows the parameters $c$ and $\phi$, as well as the distribution of $\epsilon$.
\begin{align}
Mean(X_t) & = Mean(c + \phi \, X_{t-1} + \epsilon) \\
& = Mean(c) + Mean(\phi \, X_{t-1}) + Mean(\epsilon) \\
& = c + \phi \, Mean(X_{t-1}) + 0 \\
\end{align}
Assuming that $Mean(X_{t}) = Mean(X_{t-1})$, it follows that:
\begin{align}
Mean(X_t) = \frac{c}{1 – \phi}
\end{align}
Similarly for the variance:
\begin{align}
Var(X_t) & = Var(c + \phi \, X_{t-1} + \epsilon) \\
& = Var(c) + Var(\phi \, X_{t-1}) + Var(\epsilon) \\
& = 0 + \phi^2 \, Var(X_{t-1}) + \sigma_{\epsilon}^2 \\
\end{align}
Assuming that $Var(X_{t}) = Var(X_{t-1})$, it follows that:
\begin{align}
Var(X_t) = \frac{\sigma_{\epsilon}^2}{1 – \phi^2}
\end{align}
My question
How can this be generalised to higher order moments, central moments and/or cumulants?
Following this post, is it correct to assume that:
\begin{align}
\mu_k(X_t) = \frac{\mu_k(\epsilon)}{1 – \phi^k}
\end{align}
On Wikipedia, it is noted that:
…the central limit theorem indicates that $X_t$ will be approximately normally distributed when $\phi$ is close to one.
Which makes me doubt whether the above generalisation is correct.
Secondly, $\mu_k$ refers to cumulants. Is there a similar expression for moments and/or central moments?
Context
I would like to end up with an AR(1) process which generates data with a particular non-normal distribution. I want to be able to specify the desired mean, standard deviation and skewness and subsequently work back what distribution I need for $\epsilon$ to give me the expected result.
Best Answer
Your formulas for $\mbox{Var}X_t$ and the higher order cumulants $\mu_k(X_t)$ are both correct but note that the skewness of $X_t$ (the third standardised moment) is
$$ \mbox{Skew}X_t = \frac{\mu_3(X_t)}{\mu_2(X_t)^{3/2}}=\frac{\mu_3(\epsilon_t)(1-\phi^2)^{3/2}}{\mbox{Var}(\epsilon_t)^{3/2}(1-\phi^3)}=\mbox{Skew}(\epsilon_t)\frac{(1-\phi^2)^{3/2}}{(1-\phi^3)} $$ In contrast to $\mu_3(X_t)$ this goes to zero as expected from the central limit theorem as $\phi$ goes to 1 no matter how skewed the white noise is.