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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?

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  • $\begingroup$ 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. $\endgroup$
    – whuber
    Jan 11, 2018 at 23:55

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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.

Can I use a time series method to predict a persons x against time and d?

Yes, tell us your specific model, and we can give you prediction equations, and we can then talk about what sort of errors you can expect for these particular predictions.

Since $x_t$ is related to $x_{t−1},x_{t−2}$ I would think a time series technique would be applicable.

Yes, you are probably correct. You probably need a time series model. If you want suggestions on which model to use, or information about how to pick models, you need to give us more information. I would probably put it in a new question thread, separate from this one.

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