A bit of context
I am looking for a lag-1 autoregressive process with non-Gaussian innovation/residual error, which is capable of producing both skewed and non-skewed marginal distributions.
I am aware of non-Gaussian conditional AR(1) processes (references cited in this CV answer, especially Grunwald, Hyndman, & Tedesco, 1995). Among them, the GAR(1) model of Gaver and Lewis (1980) and Lawrance (1982) is a great choice as it can produce marginal Gamma distributions. Though, the interpretation of the model is so peculiar for my target readership.
So, as an alternative, I am considering simply replacing the Gaussian i.i.d. innovations of a normal AR(1) with $\chi^2$ distribution with $k$ degrees of freedom, or more generally, Gamma distribution with shape parameter $\alpha$ and scale parameter $\lambda$:
$$X_t = c + \phi X_{t-1} + \epsilon_t, \ \ \cases{\mathbb{A}: \epsilon_t \sim \chi^2(k) \\ \mathbb{B}: \epsilon_t \sim \Gamma(\alpha, \lambda)}$$
What I am looking for
I am looking for analytical expressions for the (approximate) marginal mean and skewness of an AR(1) process with either of these distributions. (Variance is not super important to me.)
(I know the $\chi^2$ distribution is a special case of the Gamma distribution. Though in case the results are hard to attain with Gamma innovations, I can live with results for the $\chi^2$ innovations.)
What I already know
- I know one can write the AR(1) as an infinite-order moving average model, and deriving the marginal distribution via the weighted sum of the innovations:
$$X_t = \mu + \sum_{l=0}^{\infty} \phi^l \epsilon_{t-l}$$
I know one can derive the moment generating function of weighted sums of Gamma-distributed random variables of different shapes ($\alpha_i$) but the same scale ($\lambda$), which is expressed in Di Salvo (2008), which is a quite complicated, and I do not know how to simplify it for the case of the infinite sum of exponentially decaying random variables (given the $MA(\infty$) formulation above.
Mathai (1982, pp. 591-592) has mentioned that a similar summation has been studied by others and only cites Prabhu (1965), which I could not find it online:
Any ideas on how to derive the mean and skewness of the marginal distribution in either case?
References
Di Salvo, F. (2008). A characterization of the distribution of a weighted sum of gamma variables through multiple hypergeometric functions. Integral Transforms and Special Functions, 19(8), 563–575. https://sci-hub.se/10.1080/10652460802045258
Gaver, D. P., & Lewis, P. A. W. (1980). First-Order Autoregressive Gamma Sequences and Point Processes. Advances in Applied Probability, 12(3), 727–745. https://sci-hub.se/10.2307/1426429
Grunwald, G. K., Hyndman, R. J., & Tedesco, L. M. (1995). A unified view of linear AR(1) models. http://robjhyndman.com/papers/ar1.pdf
Lawrance, A. J. (1982). The Innovation Distribution of a Gamma Distributed Autoregressive Process. Scandinavian Journal of Statistics, 9(4), 234–236. https://sci-hub.se/10.2307/4615888
Mathai, A. M. (1982). Storage capacity of a dam with gamma type inputs. Annals of the Institute of Statistical Mathematics, 34(3), 591–597. https://sci-hub.se/10/c75ggp