Law of large numbers
Forms

There are two different versions of the law of large numbers that are described below. They are called the strong law of large numbers and the weak law of large numbers. Stated for the case where X, X,... is an infinite sequence of independent and identically distributed (i.i.d.) Lebesgue integrable random variables with expected value E(X) = E(X) =... = μ, both versions of the law state that the sample average

X ¯ n = 1 n (X 1 + ⋯ + X n) {\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}

converges to the expected value:

(Lebesgue integrability of X means that the expected value E(X) exists according to Lebesgue integration and is finite. It does not mean that the associated probability measure is absolutely continuous with respect to Lebesgue measure.)

Introductory probability texts often additionally assume identical finite variance Var (X i) = σ 2 {\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} (for all i {\displaystyle i}) and no correlation between random variables. In that case, the variance of the average of n random variables is

Var (X ¯ n) = Var (1 n (X 1 + ⋯ + X n)) = 1 n 2 Var (X 1 + ⋯ + X n) = n σ 2 n 2 = σ 2 n. {\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}

which can be used to shorten and simplify the proofs. This assumption of finite variance is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway.

Mutual independence of the random variables can be replaced by pairwise independence or exchangeability in both versions of the law.

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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