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I was wondering what is a good source / reference to study probability theory with proofs of the CLT and Berry-Esseen Theorems? I am pretty well-versed in the foundations of measure theory but less so in probability (i.e. stuff like characteristic functions, moments, etc) so if there's a text that suits my specific pre-requisite well please let me know, thanks!

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Most graduate level texts on probability will prove some flavor of CLT. There is one text I know of that also proves Berry-Esseen:

“Probability Theory” by Alexander Borokov

https://www.springer.com/cda/content/document/cda_downloaddocument/9781447152002-t1.pdf?SGWID=0-0-45-1403085-p175128623

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