# On the existence of the probability density of stochastic processes of the type $X_t = \sum_{i = 0}^{\infty} \phi^i W_{i-t}$.

Let the stochastic process $$\{X_t \}_{t \in \mathbb{N}}$$ be defined as $$X_t = \sum_{i = 0}^{\infty} \phi^i W_{i-t}$$

where $$\phi \in \mathbb{R}$$ and the $$W_i$$ are white noise with $$W_i\sim N(0, \sigma) \forall i \in \mathbb{Z}$$. It is well known in econometric theory that these kind of processes are solutions to auto-regressive (and ARMA) type equations.

It is quite natural to wonder if the associated distribution function to the process

$$F_t(x) = P(X_t \le x) \quad x \in \mathbb{R}$$

has a density (this type of question is asked all the time in the theory of stochastic differential equations, that is when we have continuous time $$t$$). So do we have a density in this case and can it be recovered explicitly?

• Is the following correct? For $|\phi| < 1$, $X_t = W_{-t}+\phi W_{1-t}+\ldots \sim \mathcal{N}\left(0,\sigma^2\sum_{i=0}^\infty \phi^{2i}\right) = \mathcal{N}\left(0,\frac{\sigma^2}{1-\phi^2}\right)$ – Timothy Hedgeworth Feb 28 at 17:24

To expand Hedgeworth's comments, we need two premises to hold in order to have $$X_t \sim N(0,\sigma^2 \sum_{i=0}^\infty \phi^{2i})$$ (Note that I use $$\sigma^2$$ to represent the variance of $$W_i$$).
We can easily prove the second premise by calculating that the $$cov(\phi^i W_{i-t}, \phi^j W_{j-t})=0$$.
The independence is a bit less obvious. Given your wording of the problem, my understanding is that the white noises are independent across the elements in the summation because you allow different subscripts for $$W_{i-t}$$ over different i.
So both premises hold and $$X_t \sim N(0,\sigma^2 \sum_{i=0}^\infty \phi^{2i})$$