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Central Limit Theorem(s)

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At some point, I want to collect all the ways of saying why we should expect Gaussian fluctuations in the large- 𝑛 limit when everything is nice, and why we should expect other limiting distributions in other cases... A partial, off-the-top-of-my-head list: Things I need to think about: permanent link for this note RSS feed for this note

Central Limit Theorem(s) Notebooks Central Limit Theorem(s) Last update : 07 Jul 2025 13:49 First version : 21 October 2024 \[ \newcommand{\Prob}[1]{\mathbb{P}\left( #1 \right)} \] At some point, I want to collect all the ways of saying why we should expect Gaussian fluctuations in the large-$n$ limit when everything is nice, and why we should expect other limiting distributions in other cases... A partial, off-the-top-of-my-head list: The cumulant-generating-function argument: cumulant generating functions add for independent random variables, so the CGF of an (appropriately scaled and center

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