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I've been reading my GARCH class notes and I found some discussions about stationarity.

"In Finance, stationarity of order 2 is often considered as more restrictive than the strict stationarity, because the financial returns often display distributions with fat tails (and thus only low-order moments are likely to exist)."

I don't see why stationarity of order 2 is more restrictive than strict stationarity ?? I mean strict stationarity implies the joint distribution of any moments (including moments of order two) of any degree within the process is never dependent on time.

Richard Hardy
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Ayoub McCruise
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    Does this answer your question? [A Strict Sense Stationary (SSS) process implies it is a Wide Sense Stationary (WSS) process - proof](https://stats.stackexchange.com/questions/409389/a-strict-sense-stationary-sss-process-implies-it-is-a-wide-sense-stationary-w) – Richard Hardy May 26 '20 at 10:53
  • Some relevant alternatives: [1](https://stats.stackexchange.com/questions/275783/), [2](https://stats.stackexchange.com/questions/65353/), [3](https://stats.stackexchange.com/questions/50261/). – Richard Hardy May 26 '20 at 11:00
  • I still don't see why they say that stationarity of order 2 is more restrictive than strict stationarity. Shouldn't be the other way around ? – Ayoub McCruise May 26 '20 at 19:35
  • I hope it is clear that (1) strict stationarity does not imply wide stationarity and (2) wide stationarity does not imply strict stationarity. Then in which sense is one more restrictive than the other? – Richard Hardy May 26 '20 at 19:41

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