So a time series assumes a stationary process. So, if the mean $\mu$ and the variance $\sigma^2$ are fixed for all $t$, what is the point?

Isn't the next one is just going to come out of $N(\mu, \sigma)$? What is there to predict?

Suppose I have data about how a person's measurement $x$ varies against $d$, the number of minutes until a deadline. This is not stationary. Can I use a time series method to predict a persons $x$ against time and $d$?

Since $x_t$ is related to $x_{t-1}, x_{t-2}$ I would think a time series technique would be applicable. If not, what technique would I use?

• This comment concerns your preliminary questions. Imagine a process in which you observe a random variable--maybe it has a mean of $\mu$ and variance $\sigma^2$, but you don't know them--at time $0$. From that point on, at times $t=1,2,\ldots,$ instead of re-observing that variable, you simply copy down the original value. This is a stationary time series in which every observation comes from a $N(\mu,\sigma)$ distribution. If you share that original observation with me, I can predict all subsequent observations perfectly. This is an extreme example of what prediction can accomplish.
– whuber
Commented Jan 11, 2018 at 23:55

The variance and mean you are talking about do not refer to any sort of relationship between random variables through time. For example, we could have an AR(1) process where $E[X_t \mid X_{t-1} = x_{t-1}] = \mu + \phi (x_{t-1} - \mu)$, but at the same time the unconditional mean is $E[X_t] = \mu$. You can predict with either of these, but you will have lower forecasting error if you predict with the true model's conditional expectation, taking into account the relevant information.
Since $x_t$ is related to $x_{t−1},x_{t−2}$ I would think a time series technique would be applicable.