Forms
Two different versions of the Law of Large Numbers are described below; they are called the Strong Law of Large Numbers, and the Weak Law of Large Numbers. Both versions of the law state that – with virtual certainty – the sample average
converges to the expected value
where X1, X2, ... is an infinite sequence of i.i.d. integrable random variables with expected value E(X1) = E(X2) = ...= µ. Integrability means that E(|Xj|) < ∞ for j=1,2,....
An assumption of finite variance Var(X1) = Var(X2) = ... = σ2 < ∞ is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway. This assumption is often used because it makes the proofs easier and shorter.
The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables.
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