In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated in the 1920s. In statistics, the CLT can be stated as: let X 1 , X 2 , … , X n {\displaystyle X_{1},X_{2},\dots ,X_{n}} denote a statistical sample of size n {\displaystyle n} from a population with expected value (average) μ {\displaystyle \mu } and finite positive variance σ 2 {\displaystyle \sigma ^{2}} , and let X ¯ n {\displaystyle {\bar {X}}_{n}} denote the sample mean (which is itself a random variable). Then the limit as n → ∞ {\displaystyle n\to \infty } of the distribution of ( X ¯ n − μ ) n {\displaystyle ({\bar {X}}_{n}-\mu ){\sqrt {n}}} is a normal distribution with mean 0 {\displaystyle 0} and variance σ 2 {\displaystyle \sigma ^{2}} . In other words, suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average (arithmetic mean) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the probability distribution of these averages will closely approximate a normal distribution. The central limit theorem has several variants. In its common form, the random variables must be independent and identically distributed (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions. The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem.
Independent sequences
Classical CLT Let ( X n ) n ≥ 1 {\displaystyle (X_{n})_{n\geq 1}} 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 normalized mean n ( X ¯ n − μ ) {\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )} , i.e. the difference between the sample average X ¯ n {\displaystyle {\bar {X}}_{n}} and its limit μ , {\displaystyle \mu ,} scaled by the factor n {\displaystyle {\sqrt {n}}} , 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:
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 [ n ( X ¯ n − μ ) ≤ z ] = lim n → ∞ P [ n ( X ¯ n − μ ) σ ≤ z σ ] = Φ ( z σ ) , {\displaystyle \lim _{n\to \infty }\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]=\lim _{n\to \infty }\mathbb {P} \left[{\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq {\frac {z}{\sigma }}\right]=\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 [ n ( X ¯ n − μ ) ≤ z ] − Φ ( z σ ) | = 0 , {\displaystyle \lim _{n\to \infty }\;\sup _{z\in \mathbb {R} }\;\left|\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]-\Phi \left({\frac {z}{\sigma }}\right)\right|=0~,}
where sup {\displaystyle \sup } denotes the supremum (i.e. least upper bound) of the set.
Lyapunov CLT In this variant of the central limit theorem the random variables X i {\textstyle X_{i}} have to be independent, but not necessarily identically distributed. The theorem also requires that random variables | X i | {\textstyle \left|X_{i}\right|} have moments of some order ( 2 + δ ) {\textstyle (2+\delta )} , and that the rate of growth of these moments is limited by the Lyapunov condition given below.
In practice it is usually easiest to check Lyapunov's condition for δ = 1 {\textstyle \delta =1} . If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.
Lindeberg (-Feller) CLT
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from Lindeberg in 1920). Suppose that for every ε > 0 {\textstyle \varepsilon >0} ,
lim n → ∞ 1 s n 2 ∑ i = 1 n E [ ( X i − μ i ) 2 ⋅ 1 { | X i − μ i | > ε s n } ] = 0 {\displaystyle \lim _{n\to \infty }{\frac {1}{s_{n}^{2}}}\sum _{i=1}^{n}\operatorname {E} \left[(X_{i}-\mu _{i})^{2}\cdot \mathbf {1} _{\left\{\left|X_{i}-\mu _{i}\right|>\varepsilon s_{n}\right\}}\right]=0}
where 1 { … } {\textstyle \mathbf {1} _{\{\ldots \}}} is the indicator function. Then the distribution of the standardized sums
1 s n ∑ i = 1 n ( X i − μ i ) {\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)}
converges towards the standard normal distribution N ( 0 , 1 ) {\textstyle {\mathcal {N}}(0,1)} .
CLT for the sum of a random number of random variables Rather than summing an integer number n {\displaystyle n} of random variables and taking n → ∞ {\displaystyle n\to \infty } , the sum can be of a random number N {\displaystyle N} of random variables, with conditions on N {\displaystyle N} . For example, the following theorem is Corollary 4 of Robbins (1948). It assumes that N {\displaystyle N} is asymptotically normal (Robbins also developed other conditions that lead to the same result).
Multidimensional CLT Proofs that use characteristic functions can be extended to cases where each individual X i {\textstyle \mathbf {X} _{i}} is a random vector in R k {\textstyle \mathbb {R} ^{k}} , with mean vector μ = E [ X i ] {\textstyle {\boldsymbol {\mu }}=\operatorname {E} [\mathbf {X} _{i}]} and covariance matrix Σ {\textstyle \mathbf {\Sigma } } (among the components of the vector), and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution. Summation of these vectors is done component-wise. For i = 1 , 2 , 3 , … , {\displaystyle i=1,2,3,\ldots ,} let
X i = [ X i ( 1 ) ⋮ X i ( k ) ] {\displaystyle \mathbf {X} _{i}={\begin{bmatrix}X_{i}^{(1)}\\\vdots \\X_{i}^{(k)}\end{bmatrix}}}
be independent random vectors. The sum of the random vectors X 1 , … , X n {\displaystyle \mathbf {X} _{1},\ldots ,\mathbf {X} _{n}} is
∑ i = 1 n X i = [ X 1 ( 1 ) ⋮ X 1 ( k ) ] + [ X 2 ( 1 ) ⋮ X 2 ( k ) ] + ⋯ + [ X n ( 1 ) ⋮ X n ( k ) ] = [ ∑ i = 1 n X i ( 1 ) ⋮ ∑ i = 1 n X i ( k ) ] {\displaystyle \sum _{i=1}^{n}\mathbf {X} _{i}={\begin{bmatrix}X_{1}^{(1)}\\\vdots \\X_{1}^{(k)}\end{bmatrix}}+{\begin{bmatrix}X_{2}^{(1)}\\\vdots \\X_{2}^{(k)}\end{bmatrix}}+\cdots +{\begin{bmatrix}X_{n}^{(1)}\\\vdots \\X_{n}^{(k)}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}X_{i}^{(1)}\\\vdots \\\sum _{i=1}^{n}X_{i}^{(k)}\end{bmatrix}}}
and their average is
X ¯ n = [ X ¯ i ( 1 ) ⋮ X ¯ i ( k ) ] = 1 n ∑ i = 1 n X i . {\displaystyle \mathbf {{\bar {X}}_{n}} ={\begin{bmatrix}{\bar {X}}_{i}^{(1)}\\\vdots \\{\bar {X}}_{i}^{(k)}\end{bmatrix}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {X} _{i}.}
Therefore,
1 n ∑ i = 1 n [ X i − E ( X i ) ] = 1 n ∑ i = 1 n ( X i − μ ) = n ( X ¯ n − μ ) . {\displaystyle {\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}\left[\mathbf {X} _{i}-\operatorname {E} \left(\mathbf {X} _{i}\right)\right]={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}(\mathbf {X} _{i}-{\boldsymbol {\mu }})={\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right).}
The multivariate central limit theorem states that
n ( X ¯ n − μ ) ⟶ d N k ( 0 , Σ ) , {\displaystyle {\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}_{k}(0,{\boldsymbol {\Sigma }}),}
where the covariance matrix Σ {\displaystyle {\boldsymbol {\Sigma }}} is equal to
Σ = [ Var ( X 1 ( 1 ) ) Cov ( X 1 ( 1 ) , X 1 ( 2 ) ) Cov ( X 1 ( 1 ) , X 1 ( 3 ) ) ⋯ Cov ( X 1 ( 1 ) , X 1 ( k ) ) Cov ( X 1 ( 2 ) , X 1 ( 1 ) ) Var ( X 1 ( 2 ) ) Cov ( X 1 ( 2 ) , X 1 ( 3 ) ) ⋯ Cov ( X 1 ( 2 ) , X 1 ( k ) ) Cov ( X 1 ( 3 ) , X 1 ( 1 ) ) Cov ( X 1 ( 3 ) , X 1 ( 2 ) ) Var ( X 1 ( 3 ) ) ⋯ Cov ( X 1 ( 3 ) , X 1 ( k ) ) ⋮ ⋮ ⋮ ⋱ ⋮ Cov ( X 1 ( k ) , X 1 ( 1 ) ) Cov ( X 1 ( k )