| Version 3 |
Version 2 |
| A sequence of random variables $X_1, X_2,\dots$ with finite expectations |
A sequence of random variables $X_1, X_2,\dots$ with finite expectations |
| in a probability space is said to satisfiy the strong law of large numbers |
in a probability space is said to satisfiy the strong law of large numbers |
| $$ \frac{1}{n}\left(\sum_{k=1}^n (X_k -\operatorname{E}X_k) \xrightarrow[]{a.s.} 0, $$ |
$$ \frac{1}{n}\left(\sum_{k=1}^n X_k - \sum_{k=1}^n |
|
\operatorname{E}X_k \right) \xrightarrow[]{a.s.} 0, $$ |
| where $a.s.$ stands for almost sure convergence. |
where $a.s.$ stands for almost sure convergence. |
| When the random variables are indentically distributed, with expectation $\mu$, |
When the random variables are indentically distributed, with expectation $\mu$, |
| the law becomes: |
the law becomes: |
| $$ \frac{1}{n}\sum_{k=1}^n X_k\xrightarrow[]{a.s.} \mu.$$ |
$$ \frac{1}{n}\sum_{k=1}^n X_k\xrightarrow[]{a.s.} \mu.$$ |
| Kolmogorov's strong law of large numbers theorems give conditions on the random variables under wich the law is satisfied. |
Kolmogorov's strong law of large numbers theorems give conditions on the random variables under wich the law is satisfied. |
