Chapter 4_Asymptotic Theory.pdf

Ch. 4 Asymptotic Theory
From the discussion of last Chapter it is obvious that determining the dis-
tribution of h(X1; X2; :::; XT ) is by no means a trival exercise. It turns out that
more often than not we cannot determine the distribution exactly. Because of the
importance of the problem, however, we are forced to develop approximations;
the subject of this Chapter.
This Chaper will cover the limit theorem. The terms ’limit theorems’ refers
to several theorems in probability theory under the generic names, ’laws of large
numbers’(LLN) and ’central limit theorem’(CLT). These limit theorem consis-
tute one of the most important and elegent chapters of probability theory and
play a crucial roal in statistical inferences.
1 Consistency
In this section we introduce the concepts needed to analyzed the behaviors of a
random variable indexed by the size of a sample, say
ˆ, as T .
T →∞
Limits
Definition:
Let b T , or just b be a sequence of real numbers. If there exists a real num-
{ T }1 { T }
ber b and if for every
> 0 there exist an integer N(
) such that for all T N,
≥
b b <
, then b is the limit of the sequence b .
| T −| { T }
In this definition the constant
can take on any real value, but it is the very
small values of
that provide the definition with its impact. By choosing a very
small
, we ensure that bT gets arbitrarily close to its limit b for all T that are suf-
ficiently large. When a limit exists, we say that the sequence b converges to b
{ T }
as T tends to infinity, written as b b as T . We also write b = lim b .
T →→∞ T →∞ T
When no ambiguity is possible, we simply write b b or b = lim b .
T → T
Example:
Let
2T ( 1)T
a = −−:
T 2T
1
Here 1 = limT →∞ aT , for
2T ( 1)T 1
a 1 = −− 1 = :
| T −| 2T − 2T
 
 
 
Since by binomial theorem we ha ve 
T (T + 1)
2T = (1 + 1)T = 1 + T + +1 > T:
2 · · ·
Hence, if we choose N = 1=
or large, we have, for T > N,
1 1 1
a 1 = < <
: