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Calculating the variance

Let S be the sum of n random variables. Many central limit theorems provide conditions such that S / √ Var(S) converges in distribution to N (0,1) (the normal distribution with mean 0, variance 1) as n → ∞. In some cases, it is possible to find a constant σ and function f(n) such that S /(σ √ n⋅f (n)) converges in distribution to N (0,1) as n → ∞.

Lemma — Suppose X 1, X 2, … {\displaystyle X_{1},X_{2},\dots } is a sequence of real-valued and strictly stationary random variables with E (X i) = 0 {\displaystyle \operatorname {E} (X_{i})=0} for all i {\displaystyle i}, g: → R {\displaystyle g:\to \mathbb {R} }, and S n = ∑ i = 1 n g (i n) X i {\displaystyle S_{n}=\sum _{i=1}^{n}g\left({\tfrac {i}{n}}\right)X_{i}}. Construct σ 2 = E (X 1 2) + 2 ∑ i = 1 ∞ E (X 1 X 1 + i) {\displaystyle \sigma ^{2}=\operatorname {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}