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When I was looking at time series I noticed that a common approach to time series modelling is the ARIMA model which basically does differencing until a stationary series is found and then fits the ARMA parameters.

Is it possible to construct time series that are infinitely often differentiable but never stationary when differenced?

I did not see how to construct one and was thinking of a series similar to the Weierstrass never differentiable function but this did not seem fruitful.

Ideas would be appreciated!

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    $\begingroup$ How about a time series that monotonically increases through time? For exp. one generated from the exponential function? $\endgroup$ – user2974951 Oct 17 '18 at 11:51
  • $\begingroup$ Yes I was also thinking about the exponential function as it is infintely often differentiable. $\endgroup$ – Jan Oct 17 '18 at 11:56
  • $\begingroup$ I was hoping to also know if such a series could exist if the series is bounded $\endgroup$ – Jan Oct 17 '18 at 11:58
  • $\begingroup$ Relevant/related: stats.stackexchange.com/questions/180270/… $\endgroup$ – Glen_b Oct 17 '18 at 13:09
  • $\begingroup$ Thanks for the link @Glen_b. It is an interesting post and somewhat relevant, however that post is about a time series example that has no unit root but is non-stationary. In my question we can have as many unit roots as we want, I am looking for a series that never becomes sationary when differencing. This is not tackled in the post you linked and thus deserves a new question! $\endgroup$ – Jan Oct 17 '18 at 13:17

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