Theory of Statistics - George Mason University

Multivariate Asymptotic Expectation
1.4 Limit Theorems 101
The multivariate generalization of asymptotic expectation is straightforward:
Let {Xn} be a sequence of random k-vectors, and let {An} be a sequence of
k×k positive definite matrices such that either limn→∞ An diverges (that is, in
the limit has no negative diagonal elements and some diagonal elements that
are positively infinite) or else limn→∞ An = A, where A is positive definite
and such that AnXn d → X, with E(|X|) < ∞. Then an asymptotic expectation
of {Xn} is E(A −1
n X).
If the asymptotic expectation of {Xn} is B(n)µ for some matrix B(n), and
g is a Borel function from IR k to IR k that is differentiable at µ, then by Theorem
1.46 on page 93 the asymptotic expectation of {g(Xn)} is B(n)Jg(µ)) T µ.
1.4 Limit Theorems
We are interested in functions of a sequence of random variables {Xi | i =
1, . . ., n}, as n increases without bound. The functions of interest involve either
sums or extreme order statistics. There are three general types of important
limit theorems: laws of large numbers, central limit theorems, and extreme
value theorems.
Laws of large numbers give limits for probabilities or for expectations of
sequences of random variables. The convergence to the limits may be weak or
strong.
Historically, the first versions of both laws of large numbers and central
limit theorems applied to sequences of binomial random variables.
Central limit theorems and extreme value theorems provide weak convergence
results, but they do even more; they specify a limiting distribution.
Central limit theorems specify a limiting infinitely divisible distribution, often
a normal distribution; and extreme value theorems specify a limiting extreme
value distribution, one of which we encountered in Example 1.29.
The functions of the sequence of interest are of the form
n
(1.204)
or
an
i=1
Xi − bn
an X(n:n) − bn , (1.205)
where {an} is a sequence of positive real constants and {bn} is a sequence
of real constants. The sequence of normalizing constants {an} for either case
often have the form an = n −p for some fixed p > 0.
For both laws of large numbers and central limit theorems, we will be
interested in a function of the form of expression (1.204), whereas for the
extreme value theorems, we will be interested in a function of the form of
expression (1.205). An extreme value theorem, of course, may involve X(1:n)
Theory of Statistics c○2000–2013 James E. Gentle