Learnability of a finite series

Question

How could one approach the following question:

Given a series $ \{a_n\} \subset \mathbb{R}^2 $ where $n \in [N]$ , I want to characterize the predictability or learnability of it. One definition could be how easy is it to predict the next element in the sequence given the the first k elements.

For instance, Rademacher complexity and other concept seems not adequate in this case (i.e. doesn't consider the order). I want to be able to provide a number for this series, as easily as possible.

Even better is if we assume it is a stochastic function $X(t)$ , $t \in [N]$.

Answer 1

Is there some dependence of one value in the sequence on previous values? In other words, if you can treat this as a time series, there are a whole host of methods to deal with predictions, starting with SARIMAX methods (https://www.statsmodels.org/stable/generated/statsmodels.tsa.statespace.sarimax.SARIMAX.html https://towardsdatascience.com/time-series-forecasting-with-arima-sarima-and-sarimax-ee61099e78f6 and working up to recurrent neural networks and deep learning. If your series is truly stochastic, expect it to be challenging :-p