I am currently reading some lectures I found online and which seemed really good written until I came upon Pearson's theorem ( http://studylib.net/doc/13587345/lecture-23-23.1-pearson%E2%80%99s-theorem ) (the link of the file) where the author on pg 91-92 (i.e. 2-3 in pdf) claims that the two joint distributions are equal which seems to follow from a the equality of the marginal distributions and the same covariance structure (even thought it is not obvious that the $Z_i$ have even a multi normal joint distribution). I have tried to prove this part myself for days now, and I was not successful, I have also asked here a few questions that would bring some of my attempts to prove this fact to an end but nothing again. I would be really thankful if somebody could help me understand this part.
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4 What a lot of huffing and puffing. Of course the count vectors (suitably recentered and and rescaled) have a multidimensional joint Gaussian limit, from which the result can be read off without trouble. Presumably the course has stated a scalar form of the central limit theorem but not a vector form, which leads to the tangle the author created here.
- $\begingroup$ I really do not understand what you are saying, can you explain why the vector of $Z_i$'s has a multidimensional joint Gaussian distribution, this would be the end of my problems with this theorem. $\endgroup$measuretheory– measuretheory2017-08-06 16:14:46 +00:00Commented Aug 6, 2017 at 16:14
- $\begingroup$ The count vector it not itself Gaussian, but it has a Gaussian limiting distribution. I don't know how much you know about limiting distributions so don't know how to pitch my answer. Do you know a proof for the scalar central limit theorem, and the relevance of the characteristic function? If so, the vector CLT is easy to think about. $\endgroup$kimchi lover– kimchi lover2017-08-06 16:31:01 +00:00Commented Aug 6, 2017 at 16:31
- $\begingroup$ I think I see where you are going with this, you want to say that instead of using the CLT for each of the components we use multivariate CLT to obtain a convergence to a multivariate normal distribution where the $Z_i$'s are only marginals from the multivariate normal that is obtained from the CLT? Please correct me if I am wrong. $\endgroup$measuretheory– measuretheory2017-08-06 16:55:27 +00:00Commented Aug 6, 2017 at 16:55
- $\begingroup$ That's it exactly! $\endgroup$kimchi lover– kimchi lover2017-08-06 17:04:54 +00:00Commented Aug 6, 2017 at 17:04