Approximation starts here

Sample mean

Theorem 6.2.4 Law of Large Numbers. Suppose that X1,...,Xn form a random sample from a distribution for which the mean is μ and for which the variance is finite. Let Xn denote the sample mean. Then

Theorem 6.3.1 Central Limit Theorem (Lindeberg and L´evy). If the random variables X1,...,Xn form a random sample of size n from a given distribution with mean μ and variance σ2 (0 < σ2 < ∞), then for each fixed number x,

where Phi denotes the c.d.f. of the standard normal distribution.

Functions of the sample mean

Theorem 6.3.2 Delta Method. Let Y1, Y2,... be a sequence of random variables, and let F∗ be a continuous c.d.f. Let θ be a real number, and let a1, a2,... be a sequence of positive numbers that increase to ∞. Suppose that an(Yn − θ ) converges in distribution to F∗. Let α be a function with continuous derivative such that α'(θ ) not= 0. Then an[α(Yn) −α(θ )]/α'(θ ) converges in distribution to F∗.

Here Yn is not specified. 

Corollary 6.3.1 Delta Method for Average of a Random Sample. Let X1, X2,... be a sequence of i.i.d. random variables from a distribution with mean μ and fifinite variance σ2. Let α be a function with continuous derivative such that α�(μ) = 0. Then the asymptotic distribution of

is the standard normal distribution.

amount to Yn is the sample mean