Central limit theorem

The central limit theorem states that sums or averages of many independent random variables, after suitable normalization, converge in distribution to a normal distribution.

Convergence in distribution: A mode of convergence in which random variables’ cumulative distribution functions converge at continuity points of the limiting distribution. The theorem’s conclusion is convergence in distribution, not necessarily convergence of individual outcomes.
Random variable: A measurable quantity whose value is determined by the outcome of a random process. The theorem applies to sequences of random variables, not directly to arbitrary datasets.
Sampling distribution: The probability distribution of a statistic computed from repeated samples. The theorem explains why many sample means have approximately normal sampling distributions.
Berry–Esseen theorem: A theorem that bounds the error in approximating a normalized sum’s distribution by the normal distribution. It quantifies the rate at which a classical central limit approximation improves.
Standardization: A transformation that shifts a variable by its mean and scales it by its standard deviation. Centering and scaling put sums into the form whose limiting distribution is standard normal.
Expected value: The probability-weighted average value of a random variable. The classical theorem centers each summand by subtracting its expected value.
Standard error: The standard deviation of a statistic’s sampling distribution. For sample means, the theorem motivates the familiar standard-error scale.
Lyapunov central limit theorem: A central limit theorem for independent, potentially non-identically distributed variables under a moment condition. It extends the classical result beyond identically distributed summands.
Variance: A measure of how widely a random variable’s values are dispersed around its mean. The classical normalization uses variance to set the scale of a sum.
Identically distributed random variables: Random variables that share the same probability distribution. The simplest classical version assumes every summand has the same distribution.

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