strong law of large numbersStrongLawOfLargeNumbers2002-12-08 03:45:302006-09-28 02:50:47Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA sequence of random variables $X_1, X_2,\dots$ with finite expectations
in a probability space is said to satisfiy the \textit{strong law of large numbers} if
$$ \frac{1}{n}\sum_{k=1}^n (X_k -\operatorname{E}[X_k]) \xrightarrow[]{a.s.} 0, $$
where $a.s.$ stands for convergence almost surely.
When the random variables are identically distributed, with expectation $\mu$,
the law becomes:
$$ \frac{1}{n}\sum_{k=1}^n X_k\xrightarrow[]{a.s.} \mu.$$
Kolmogorov's strong law of large numbers theorems give conditions on the random variables under which the law is satisfied.