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strong law of large numbers (Definition)

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 if

$\displaystyle \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:

$\displaystyle \frac{1}{n}\sum_{k=1}^n X_k\xrightarrow[]{a.s.} \mu.$

Kolmogorov's strong law of large numbers theorems give conditions on the random variables under which the law is satisfied.



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Cross-references: Kolmogorov's strong law of large numbers, identically distributed, almost surely, numbers, probability space, expectations, finite, random variables, sequence
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This is version 8 of strong law of large numbers, born on 2002-12-08, modified 2006-09-28.
Object id is 3685, canonical name is StrongLawOfLargeNumbers.
Accessed 6848 times total.

Classification:
AMS MSC60F15 (Probability theory and stochastic processes :: Limit theorems :: Strong theorems)

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Is this correct? by aparnack on 2006-08-17 12:19:34
I think this definition of strong law is incorrect.
Refer this link
http://en.wikipedia.org/wiki/Law_of_large_numbers
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