Let $(X_n)$ be a sequence of real, square integrable random variables. Define $S_n=\sum_{j=1}^nX_j$ and suppose $E(X_n)=0$ and $\sigma_n^2=$Var$(S_n)>0$. Conditions for $\frac{S_n}{\sigma_n}\rightarrow N(0,1)$ stably are provided. The main of such conditions is that $E(X_{n+1}\mid X_1,\ldots,X_n)\rightarrow 0$ (in some sense) at a suitable rate. Various examples are given as well.
Pratelli, L., Rigo, P. (2026). Classical central limit theorem via conditional expectations. STATISTICS & PROBABILITY LETTERS, 234(July), 1-5 [10.1016/j.spl.2026.110689].
Classical central limit theorem via conditional expectations
Rigo Pietro
2026
Abstract
Let $(X_n)$ be a sequence of real, square integrable random variables. Define $S_n=\sum_{j=1}^nX_j$ and suppose $E(X_n)=0$ and $\sigma_n^2=$Var$(S_n)>0$. Conditions for $\frac{S_n}{\sigma_n}\rightarrow N(0,1)$ stably are provided. The main of such conditions is that $E(X_{n+1}\mid X_1,\ldots,X_n)\rightarrow 0$ (in some sense) at a suitable rate. Various examples are given as well.File in questo prodotto:
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