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

Question

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?

Answer 1

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 $W_i$).

1. The elements in the summation are independent. This premise allows a normal distribution of the sum.
2. The elements are not correlated. This premise allows the variance of the sum to be simply the sum of the variance.

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 $$ X_t \sim N(0,\sigma^2 \sum_{i=0}^\infty \phi^{2i})$$

Note that it is not common to have both the premises satisfied in a typical stochastic process in the econometric theory.