
Lecture 4
Deviation of
b
F
n
(x) from F(x) as a
random process
In the previous lecture we established that the deviation of
b
F
n
(x) from
F(x) converges to 0 uniformly in x. One would like, however, to find
how rapid this convergence is. Let us clarify this first for just one fixed
point x.
Since the variance of the difference
b
F
n
(x) −F(x) is equal to
1
n
F(x)[1 −F(x)]
(see (3.6)), the normalized difference
√
n[
b
F
n
(x) −F(x)] has a “stable”
variance:
Var
√
n
b
F
n
(x) −F(x)
= F(x)
1 −F(x)
,
which is in fact independent of n. But then what can we know about
the behavior of this normalized difference as a random variable? That
is, what can we say about the probability P{
√
n|
b
F