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The Central limit theorem (CTL) is often given similar to the entry in Wikipedia as:

Suppose ${X_1, X_2, ...}$ is a sequence of independent and identically distributed random variables with $E[X_i] = µ$ and $Var[X_i] = σ^2 < ∞$. Then as $n$ approaches infinity, the random variables $\sqrt{n}(S_n − µ)$ convergence in distribution to a normal distribution $N(0, σ^2)$

What happens if the the random variables are transformed by a function $g: X_i \mapsto Z_i$? Under what conditions does the CTL hold for the random variables $Z_i$?

I am especially interested in the cases where the $Z_i$ are elements of a Hilbert space of functions.

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The Lindeberg-Feller CLT would hold as long as the transformed sequence satisfies the Lindeberg Condition

Here is an example where the transformation does not allow CLT to hold:

$X_i \sim \mathcal{N}(0,1), \;\;g:x\mapsto \frac{1}{x}$ , as the transformed sequence lacks a mean or variance. See here.

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