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Looking for a mathematical proof which shows that a Strict Sense Stationary (SSS) process is necessarily a Wide Sense Stationary (WSS) process.

D.Cohen
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  • What do you think about the answer? If you need further clarification, you may ask for that in the comments; otherwise, you may accept the answer by clicking on the tick mark to the left as described in the [Tour](https://stats.stackexchange.com/tour) of this site. – Richard Hardy Sep 22 '19 at 08:34

1 Answers1

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There can be no proof of the statement, because the statement is false. Here is why: strict-sense stationarity does not require existence of the second moment, while wide-sense stationarity does. Therefore, every process that is strict-sense stationary but does not have a finite second moment is a counterexample to the statement. One concrete example is an i.i.d. sequence of Cauchy random variables.

Richard Hardy
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    (+1) Interestingly, in many resources it says *SSS implies WSS*: http://ece-research.unm.edu/bsanthan/ece541/station.pdf , http://isl.stanford.edu/~abbas/ee278/lect07.pdf, http://circuit.ucsd.edu/~yhk/ece250-win17/pdfs/lect09.pdf. The only source stating o/w I could find is https://nptel.ac.in/courses/111103022/module9/lec1.pdf – gunes May 21 '19 at 12:29
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    @gunes Thank you for doing that research. I had originally supposed the distinctions among those materials would be subtle differences in definition of SSS and/or WSS, but that's not the case: *they all agree.* We can only conclude that the first three are sloppy (some seem to be quoting the others) and implicitly assume that the "technical condition" of finite variance holds in the assertion that SSS implies WSS. – whuber May 21 '19 at 12:44
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    @gunes If you had looked on stats.SE, you would have found the Cauchy counterexample in [this answer](https://stats.stackexchange.com/a/65359/6633) – Dilip Sarwate May 21 '19 at 15:58
  • @DilipSarwate I've already read, understood and upvoted your answer there, before writing out the above comment. I just recall constantly seeing resources quoting the implication of WSS given SSS w/o mentioning finiteness of the variance. – gunes May 21 '19 at 17:57