The central limit theorem for the i.i.d case states that for $X_k \sim \text{i.i.d.}(\mu, \sigma^2)$, we have that:
$$ \dfrac{\sum_{k=1}^{n} X_k - n \mu}{\sigma\sqrt n } \xrightarrow{d} \mathcal{N}(0,1)$$ or equivalently for $S_n = \sum_{k=1}^n X_k$:
$$ \dfrac{S_n - E(S_n)}{\sqrt{Var(S_n)}} \xrightarrow{d} \mathcal{N}(0,1) $$
I was reading the proof for the CLT and it seems to me that the need for writing it centered with $S_n-E(S_n)$ comes from the fact that the random variable $Z_n = \dfrac{S_n - E(S_n)}{\sqrt{Var(S_n)}}$ has zero mean and unit variance. This is convenient when considering the Taylor expansion for the characteristic function, because we'll have:
$$ \varphi_{Z_n}(t) = \left(1- \dfrac{t^2/2}{n} + o(1/n) \right)^n \to e^{t^2/2} \text{ as } n\to \infty$$
This is because the zero mean makes the term $ it E(X_1)=0$ for the Taylor expansion, thus making it simpler to evaluate the convergence as $n\to\infty$ for the characteristic function.
What I am wondering is: are there "versions" or "alternatives" for the central limit theorem for non-centered random variables? If so, what would be an example?
My goal is to understand how "flexible" I can be for convergence of random variables. I have seen some specific examples of convergence of random varaibles, as in Shiryaev's section of infinitely divisible distributions, but the formulation is again "centered". I know that the Lindeberg CLT works for non-i.i.d RV's, just independent ones, but the formulation is still in terms of the "centered" approach, only requiring the Lindeberg condition.
Thanks! :)
IID conditions can be relaxed quite a bit into certain "mix" conditions.
But the CLT, in the form of $\sqrt{n} Z_n \rightarrow N()$, in distribution, necessarily requires that $E(Z_n) \rightarrow 0$, because $Z_n \rightarrow 0$ in distribution, and thus in probability.
So it has to be "centered" (using your terminology) asymptotically.