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Classical CLT

Let { X 1, …, X n } {\displaystyle \{X_{1},\ldots,X_{n}}\} be a sequence of i.i.d. random variables having a distribution with expected value given by μ {\displaystyle \mu } and finite variance given by σ 2. {\displaystyle \sigma ^{2}.} Suppose we are interested in the sample average X ¯ n ≡ X 1 + ⋯ + X n n. {\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}.}

By the law of large numbers, the sample average converges almost surely (and therefore also converges in probability) to the expected value μ {\displaystyle \mu } as n → ∞. {\displaystyle n\to \infty.}

The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number μ {\displaystyle \mu } during this convergence. More precisely, it states that as n {\displaystyle n} gets larger, the distribution of the difference between the sample average X ¯ n {\displaystyle {\bar {X}}_{n}} and its limit μ, {\displaystyle \mu,} when multiplied by the factor n {\displaystyle {\sqrt {n}}} — that is, n (X ¯ n − μ) {\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu)} — approaches the normal distribution with mean 0 {\displaystyle 0} and variance σ 2. {\displaystyle \sigma ^{2}.} For large enough n, {\displaystyle n,} the distribution of X ¯ n {\displaystyle {\bar {X}}_{n}} gets arbitrarily close to the normal distribution with mean μ {\displaystyle \mu } and variance σ 2 / n. {\displaystyle \sigma ^{2}/n.}

The usefulness of the theorem is that the distribution of n (X ¯ n − μ) {\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu)} approaches normality regardless of the shape of the distribution of the individual X i. {\displaystyle X_{i}.} Formally, the theorem can be stated as follows:

Lindeberg–Lévy CLT — Suppose X 1, X 2, X 3 … {\displaystyle X_{1},X_{2},X_{3}\ldots } is a sequence of i.i.d. random variables with E = μ {\displaystyle \operatorname {E} =\mu } and Var = σ 2 < ∞. {\displaystyle \operatorname {Var} =\sigma ^{2}<\infty.} Then, as n {\displaystyle n} approaches infinity, the random variables n (X ¯ n − μ) {\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu)} converge in distribution to a normal N (0, σ 2) {\displaystyle {\mathcal {N}}(0,\sigma ^{2})}: n (X ¯ n − μ) ⟶ d N (0, σ 2). {\displaystyle {\sqrt {n}}\left({\bar {X}}_{n}-\mu \right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}\left(0,\sigma ^{2}\right).}

In the case σ > 0, {\displaystyle \sigma >0,} convergence in distribution means that the cumulative distribution functions of n (X ¯ n − μ) {\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu)} converge pointwise to the cdf of the N (0, σ 2) {\displaystyle {\mathcal {N}}(0,\sigma ^{2})} distribution: for every real number z, {\displaystyle z,} lim n → ∞ P = lim n → ∞ P = Φ (z σ), {\displaystyle \lim _{n\to \infty }\mathbb {P} \left=\lim _{n\to \infty }\mathbb {P} \left=\Phi \left({\frac {z}{\sigma }}\right),} where Φ (z) {\displaystyle \Phi (z)} is the standard normal cdf evaluated at z. {\displaystyle z.} The convergence is uniform in z {\displaystyle z} in the sense that lim n → ∞ sup z ∈ R | P − Φ (z σ) | = 0 , {\displaystyle \lim _{n\to \infty }\;\sup _{z\in \mathbb {R} }\;\left|\mathbb {P} \left-\Phi \left({\frac {z}{\sigma }}\right)\right|=0~,} where sup {\displaystyle \sup } denotes the least upper bound (or supremum) of the set.