Relationship between Data Size and Arima Prediction Interval Width?

Take an ARIMA(1,0,1) for simplicity: begin{equation} y_t = phi_0 + phi_1 y_{t-1} + theta_1 epsilon_{t-1} + epsilon_t. end{equation} Typically, this is estimated by maximimum likelihood which requires us to make an assumption about the distribution of $epsilon_t$. Most of the time, people pick a Gaussian distribution and impose homoskedasticity, i.e., they say $epsilon_t sim N(0,sigma)$.

For simplicity, we'll do the one-step ahead prediction interval, conditional on $(y_T, epsilon_T)$: begin{align} y_{T+1} | (y_T, epsilon_T) &sim N(mu_T, Sigma_T) \ mu_T &= E_T(y_{T+1}) = phi_0 + phi_1 y_T +theta_1 epsilon_T \ Sigma_T &= var_T(y_{T+1}) = sigma^2. end{align}

In practice, you will substitute the MLE estimates for the parameter values. Given the known distribution, you can build prediction interval. In other words, you will neglect the uncertainty due to the fact that parameters are estimated and not known. In other words, the prediction interval isn't a function of sample size, although in practice the fact that you rely on an asymptotic argument to replace parameter values with their MLE estimates does mean that it would bounce around if you added or subtracted observations from your sample.