1
Background Notes: Proof of Consistency
Economics 245A
There are several di&erent forms of uniform convergence of a sto
chastic function to a continuous nonstochastic function. Let
g
T
(
) de
note a nonnegative (to avoid the use of absolute value signs in the
de±nitions with
g
T
(
)) sequence of random variables that depend on
(that is, a sequence of stochastic functions of
).
(Almost Sure Uniform Convergence)
If
P
[ lim
T
!1
sup
2
g
T
(
) = 0] = 1
then
g
T
(
) converges to 0 almost surely uniformly in
2
². Frequently
the convergence is de±ned in terms of the sample average loglikelihood,
which we refer to here as
T
1
Q
T
(
). If
P
[ lim
T
!1
sup
2
j
T
1
Q
T
(
)
Q
(
)
j
= 0] = 1
then
T
1
Q
T
(
) converges to
Q
(
) almost surely uniformly in
2
².
(Weak Semiuniform Convergence)
If
lim
T
!1
inf
2
P
[
g
T
< "
] = 1
for any
" >
0, then
g
T
(
) converges to 0 in probability semiuniformly
in
2
². For the sample average loglikelihood, if
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 Fall '08
 Staff
 Economics, Probability theory, Uniform convergence, GT

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