Theory of Statistics - George Mason University

1.4 Limit Theorems 105
it applies to Sn that has a binomial distribution with parameters n and π
(because it is the sum of n iid Bernoullis with parameter π.
Theorem 1.56 (De Moivre Laplace Central Limit Theorem)
If Sn has a binomial distribution with parameters n and π, then
1 1
√ (Sn − nπ)
π(1 − π) n d → N(0, 1). (1.214)
This central limit theorem is a special case of the classical central limit
theorem for iid random variables with finite mean and variance.
Notice that Bernoulli’s theorem and the de Moivre Laplace central limit
theorem, which are stated in terms of binomial random variables, apply to
normalized limits of sums of Bernoulli random variables. This is the usual
form of these kinds of limit theorems; that is, they apply to normalized limits
of sums of random variables. The first generalizations apply to sums of
iid random variables, and then further generalizations apply to sums of just
independent random variables.
The Central Limit Theorem for iid Scalar Random Variables with
Finite Mean and Variance
Theorem 1.57
Let X1, X2, . . . be a sequence of independent random variables that are identically
distributed with mean µ and variance σ2 > 0. Then
n
1 1
d
√ Xi − nµ → N(0, 1). (1.215)
σ n
i=1
A proof of this uses a limit of a characteristic function and the uniqueness
of the characteristic function (see page 87).
Independent but Not Identical; Triangular Arrays
The more general central limit theorems apply to a triangular array; that is,
to a sequence of finite subsequences. The variances of the sums of the subsequences
is what is used to standardize the sequence so that it is convergent.
We define the sequence and the subsequences as follows.
Let {Xnj, j = 1, 2, . . ., kn} be independent random variables with kn → ∞
as n → ∞. We let
kn
Rn =
j=1
Xnj
represent “row sums”, as we visualize the sequence in an array:
Theory of Statistics c○2000–2013 James E. Gentle