I am working on a research topic where I need to add together two AR processes and I was wondering if the distribution of these processes is of a recognizable form/structure. More formally, if $x_t$ is a AR(p) process with characteristic polynomial $\Phi_x(u)$ and $y_t$ is a AR(q) process with characteristic polynomial $\Phi_y(u)$, then what is the structure of $z_t=x_t+y_t$?
Time Series – How to Determine the Sum of Autoregressive Processes
self-studytime series
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This question needs some ideas from random processes and some from Fourier theory.
The autocorrelation function of a (continuous-time finite-variance) stationary random process $\{X_t\colon -\infty < t < \infty\}$ is $R_X(t) = E[X_{\tau}X_{\tau+t}]$ and the spectral density $S_X(\omega)$ is the Fourier transform of $R_x(t)$. For your problem, the independence of the $\{X_t\}$ and $\{Y_t\}$ processes gives that
$$R_Z(t) = E[Z_{\tau}Z_{\tau+t}] = E[X_{\tau}Y_{\tau}X_{\tau+t}Y_{\tau+t}] = E[X_{\tau}X_{\tau+t}]E[Y_{\tau}Y_{\tau+t}]= R_X(t)R_Y(t).$$
So much so for random processes. From Fourier transform theory, we have that the transform of a product of two functions is the convolution of their Fourier transforms. If you are not familiar with this, see, for example, the last paragraph of Section 5.8 of the Wikipedia article on the Fourier transform. Thus, $$S_Z(\omega) = \int_{-\infty}^\infty S_X(\lambda)S_Y(\omega-\lambda)\, \mathrm d\lambda.\tag{1}$$
For your particular application with a discrete-time random process (a.k.a. time series), similar results apply but in $(1)$ the limits work out to be $-\pi$ and $\pi$. (You will need to know about the discrete-time Fourier transform to get to this). Note also that you might be missing a $\frac{1}{2\pi}$ factor in the result you state as wanting to prove in your question.
It might not be relevant for @MrJames anymore but my answer might help others.
Have a look at "Introduction to Time Series and Forecasting" by Brockwell and Davis (third edition). Here they say that a stationary solution of $\Phi(L)y_t=z_t$ exists and is unique if and only if $\Phi$ has no roots on the unit circle. It is causual if all roots of $\Phi$ are outside the unit circle.
Hence, causality implies stationarity but not the other way around.
Best Answer
This was studied by Granger and Morris (1976) who showed that
AR($p$) + AR($q$) = ARMA($p+q,\max(p,q)$).