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Pardon me if the question is stupid, but I don't understand how you go about proving exchangeability of a random sequence. In the case of a distribution such as the normal distribution, I get that since it is determined by the moments, it is all about checking if the moments and covariances are "exchangeable". However, I get stumped at a different example.

Assume we have a infinite sequence of binary random variables which we know are identically distributed, but not indepedent. How do we determine if the sequence is exchangeable (in the sense given at https://en.wikipedia.org/wiki/Exchangeable_random_variables) ? If I were to guess, I would say that it is something about the covariance, but I am not sure. In practice I feel like it is not feasible to check the joint distribution...

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    $\begingroup$ Usually, this is not "determined" but is assumed. There is no mathematical way to resolve your question unless additional information is supplied about that infinite sequence of variables. $\endgroup$ – whuber Apr 27 '20 at 13:07
  • $\begingroup$ I mean, that is kind of my question. What information about the ffds needs to be supplied to be able to make that conclusion? $\endgroup$ – edo Apr 27 '20 at 13:08
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    $\begingroup$ What are the "ffds"? $\endgroup$ – whuber Apr 27 '20 at 13:15
  • $\begingroup$ stats.stackexchange.com/questions/344794/… $\endgroup$ – kjetil b halvorsen Oct 6 '20 at 17:48
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Exchangeability is really a modeling assumption and not something you can test for empirically. Some aspects of it can be tested in some settings, but not more. So to make your Q meaningful, we must take it as asking what does this modeling assumption really imply.

Exchangeability is a form of symmetry, basically the symmetry that **the ordering of the sample is not informative. So it excludes things like time series ... see Independence of events in real-life data. So, a simple random sample is (finitely) exchangeable. More information can be found at Exchangeability and IID random variables, What is the intuition behind exchangeable samples under the null hypothesis? and search this site.

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