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| Title of object: |
strong law of large numbers |
| Canonical Name: |
StrongLawOfLargeNumbers |
| Type: |
Definition |
| Created on: |
2002-12-08 03:45:30.477507-05 |
| Modified on: |
2002-12-08 03:45:30.477507-05 |
| Classification: |
msc:60F15 |
Preamble:
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Content:
A sequence of random variables $X_1, X_2,\dots$ with finite means
in a probability space is said to satisfiy the strong law of large numbers
$$ \frac{\sum_{k=1}^n X_k - \sum_{k=1}^n
\operatorname{E}X_k}{n} \rightarrow^{a.s.} 0, $$
where $a.s.$ stands for almost sure convergence.
When the random variables are indentically distributed, with mean $\mu$,
the law becomes:
$$ \frac{\sum_{k=1}^n X_k}{n}\rightarrow^{a.s.} \mu.$$
Kolmogorov's strong law of large numbers theorems give conditions on the random variables under wich the law is satisfied. |
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