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'Lindeberg's central limit theorem'
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| Title of object: |
Lindeberg's central limit theorem |
| Canonical Name: |
LindebergsCentralLimitTheorem |
| Type: |
Theorem |
| Created on: |
2002-12-10 10:55:37.65497-05 |
| Modified on: |
2002-12-10 10:59:50.187263-05 |
| Classification: |
msc:60F05 |
| Synonyms: |
Lindeberg's central limit theorem=Lindeberg condition |
Preamble:
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Content:
Let $X_1, X_2,\dots$ be independent random variables with distribution functions $F_1,F_2,\dots$, respectively, such that $EX_n=\mu_n$ and $\operatorname{Var}X_n=\sigma_n^2<\infty$, with at least one $\sigma_n>0$.
Let \[S_n = X_1+\cdots+X_n\;\mbox{and}\; S_n=\sqrt{\operatorname{Var}(S_n)} =
\sqrt{\sigma_1^2+\cdots+\sigma_n^2}.\]
Then the normalized partial sums $\frac{S_n - ES_n}{s_n}$ converge
\PMlinkname{in distribution}{ConvergenceInDistribution} to a random variable with normal distribution $N(0,1)$, if the following \emph{Lindeberg condition} is satisfied:
\[\forall \varepsilon>0,\; \lim_{n\rightarrow\infty} \frac{1}{s_n^2}
\sum_{k=1}^n \int_{|x-\mu_k|>\varepsilon s_n} (x-\mu_k)^2 dF_k(x) = 0.\]
\emph{Remarks.}
1. The reciprocal to this theorem (due to Feller) is valid if the following additional requirement is satisfied: \[\max_{1\leq k\leq n} (\frac{\sigma_k^2}{s_n^2})\xrightarrow[n\rightarrow\infty]{} 0.\]
2. Lyapunov's central limit theorem is a consequence of this theorem, and provides a condition which is easier to handle (but less general.) |
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